Rotordynamic Analysis of Tapered Composite Driveshaft ......iv and the stiffness matrix of the...

Rotordynamic Analysis of Tapered Composite

Driveshaft Using Conventional and Hierarchical

Finite Element Formulations

Majed AL Muslmani

A thesis

In the Department

Of

Mechanical and Industrial Engineering

Presented in Partial Fulfillment of the Requirements

For the Degree of

Master of Applied Science (Mechanical Engineering) at

Concordia University

Montreal, Quebec, Canada

March 2013

©Majed Saleh AL Muslmani 2013

ii

CONCORDIA UNIVERSITY

SCHOOL OF GRADUATE STUDIES

This is to certify that the Thesis prepared,

By: Majed Al Muslmani

Entitled: “Rotordynamic Analysis of Tapered Composite Driveshaft Using

Conventional and Hierarchical Finite Element Formulations”

and submitted in partial fulfillment of the requirements for the Degree of

Master of Applied Science (Mechanical Engineering)

complies with the regulations of this University and meets the accepted standards with

respect to originality and quality.

Signed by the final Examining Committee:

___________________________________

Chair

Dr. M. K. Zanjani

___________________________________

Examiner

Dr. S. V. Hoa

___________________________________

Examiner

Dr. K. Galal (external)

___________________________________

Supervisor

Dr. R. Ganesan

Approved by: ___________________________________________________

Dr. S. Narayanswamy, MASc Program Director

Department of Mechanical and Industrial Engineering

___________________________________

Dean Dr. Robin Drew

Faculty of Engineering & Computer Science

Date: _____________

iii

Abstract

Rotordynamic Analysis of Tapered Composite Driveshaft Using Conventional and

Hierarchical Finite Element Formulations

Majed Al Muslmani

In the aerospace and automotive applications driveshafts are manufactured using fiber

reinforced composite materials. Compared to a conventional metallic driveshaft, a

composite driveshaft gives higher natural frequencies and critical speeds, and lower

vibration. The design of the driveshaft is dependent on its fundamental natural frequency

and its first critical speed, and tapering the driveshaft can substantially improve the

values of the natural frequency and first critical speed. In this thesis, the rotordynamic

analysis of the tapered composite driveshaft is carried out using three finite element

formulations: the conventional-Hermitian finite element formulation, the Lagrangian

finite element formulation, and the hierarchical finite element formulation. These finite

element models of the tapered composite shaft are based on Timoshenko beam theory, so

transverse shear deformation is considered. In addition, the effects of rotary inertia,

gyroscopic force, axial load, coupling due to the lamination of composite layers, and

taper angle are incorporated in the conventional-Hermitian, the Lagrangian, and the

hierarchical finite element models. The strain energy and the kinetic energy of the tapered

composite shaft are obtained, and then the equations of motion are developed using

Lagrange’s equations. Explicit expressions for the mass matrix, the gyroscopic matrix

iv

and the stiffness matrix of the tapered composite shaft are derived to perform

rotordynamic analysis. The Lagrangian beam finite element formulation has three nodes

and four degrees of freedom per each node while the conventional-Hermitian beam and

the hierarchical beam finite element formulations have two nodes. The three finite

element models are validated using the approximate solution based on the Rayleigh-Ritz

method. A comprehensive parametric study is conducted based on the finite element

models, which shows that tapering the composite driveshaft can increase considerably the

natural frequency and first critical speed, and that they have nonlinear variation with the

taper angle.

v

Acknowledgements

I would like to express my sincere gratitude to Prof. Rajamohan Ganesan for his

guidance, continuous encouragement, and valuable advices throughout this research.

Also, I would like to be grateful to him for his time and constructive comments during

writing my thesis.

In addition, I would like to acknowledge Ministry of Higher Education in Saudi Arabia

for the financial support. Special thank to my colleagues at office EV 13.167 for their

help and support.

Last but not least, I would like express my sincere appreciation to my family members

and my friends for their support and encouragement.

vi

Table of Contents

List of Figures ........................................................................................................................... x

List of Tables .......................................................................................................................... xv

Nomenclature ......................................................................................................................... xix

Chapter 1 Introduction and literature survey ............................................................................ 1

1.1 Rotordynamic analysis .............................................................................................. 1

1.2 Composite materials .................................................................................................. 2

1.3 Finite element method (FEM) ................................................................................... 3

1.4 Literature survey ....................................................................................................... 4

1.4.1 Rotordynamic analysis of conventional metallic shaft ....................................... 5

1.4.2 Rotordynamic analysis of composite shaft ......................................................... 7

1.5 Objectives of the thesis ........................................................................................... 10

1.6 Layout of the thesis ................................................................................................. 11

Chapter 2 Rotordynamic Analysis of Rotor-Bearing System Using Conventional Finite

Element Method ...................................................................................................................... 13

2.1 Introduction ............................................................................................................. 13

2.2 Coordinate system ................................................................................................... 13

2.3 Rigid disk element ................................................................................................... 17

vii

2.4 Bearing element....................................................................................................... 21

2.5 Shaft element ........................................................................................................... 22

2.5.1 Shape functions based on Euler - Bernoulli beam theory ................................ 24

2.5.2 Shape Functions based on Timoshenko Beam theory ...................................... 26

2.5.3 Energy equations .............................................................................................. 31

2.5.4 Tapered driveshaft ............................................................................................ 33

2.6 System equations of motion and analysis ............................................................... 34

2.6.1 Whirl speeds analysis ....................................................................................... 35

2.6.2 Campbell Diagram and Critical Speeds............................................................ 36

2.6.3 Steady-State Synchronous Response ................................................................ 38

2.7 Numerical Examples ............................................................................................... 39

2.8 Summary ................................................................................................................. 54

Chapter 3 Rotordynamic Analysis of Uniform Composite Shaft Using Finite Element

Method .................................................................................................................................... 55

3.1 Introduction ............................................................................................................. 55

3.2 Stress - strain relations for a composite material layer ........................................... 56

3.1 Strain-displacement relations .................................................................................. 60

3.2 Kinetic and strain energy expressions ..................................................................... 62

3.3 Conventional-Hermitian finite element formulation ............................................... 70

3.4 Numerical examples ................................................................................................ 77

3.5 Summary ............................................................................................................... 101

viii

Chapter 4 Rotordynamic Analysis of Tapered Composite Shaft Using Finite Element

Method .................................................................................................................................. 102

4.1 Introduction ........................................................................................................... 102

4.2 Stress-strain relations for tapered cylinder layer ................................................... 102

4.3 Kinetic and potential energy expressions .............................................................. 108

4.4 Finite element formulation .................................................................................... 115

4.4.1 Hierarchical composite shaft element formulation ......................................... 115

4.4.2 Lagrangian composite shaft element formulation .......................................... 123

4.4.3 Conventional-Hermitian composite shaft element formulation ..................... 130

4.5 Rayleigh – Ritz solution ........................................................................................ 141

4.6 Validation .............................................................................................................. 146

4.7 Summary ............................................................................................................... 150

Chapter 5 Parametric Study of Tapered Composite Shaft .................................................... 151

5.1 Introduction ........................................................................................................... 151

5.2 Tapered composite shaft case A ............................................................................ 152

5.3 Tapered composite shaft Case B ........................................................................... 166

5.3.1 Effect of length on natural frequencies and first critical speed ...................... 167

5.3.2 Effect of shaft diameter on natural frequencies and first critical speed ......... 172

5.3.3 Effect of fiber orientation on the natural frequencies and first critical speed 176

5.3.4 Effect of the stiffness of the bearing on the first critical speed ...................... 183

ix

5.3.5 Effect of axial load on natural frequencies and first critical speed ................ 185

5.3.6 Effect of the disk on the natural frequencies and first critical speed .............. 191

5.4 Summary ............................................................................................................... 194

Chapter 6 Conclusions, contributions, and future work ....................................................... 195

6.1 Conclusions ........................................................................................................... 195

6.2 Contributions ......................................................................................................... 197

6.3 Future work ........................................................................................................... 197

References ............................................................................................................................. 199

Appendix A Uniform Composite shaft-Conventional finite element ................................... 204

Appendix B Hierarchical shaft element formulation ............................................................ 215

Appendix C Lagrange’s interpolation formulation ............................................................... 234

Appendix D Conventional finite element ............................................................................. 248

x

List of Figures

Figure 2.1 Typical rotor-bearing system ........................................................................... 14

Figure 2.2 Cross-section rotation angles .......................................................................... 16

Figure 2.3 Typical rotating rigid disk. .............................................................................. 18

Figure 2.4 Rotating shaft supported at its ends by two bearings. .................................... 21

Figure 2.5 Typical finite shaft element ............................................................................ 23

Figure 2.6 The displacements and the rotations in the two bending planes .................... 24

Figure ‎2.7 Deformed geometry of an edge of beam under the assumption of Timoshenko

beam theory ....................................................................................................................... 27

Figure ‎2.8 Typical Campbell diagram of rotor-bearing system. ...................................... 38

Figure ‎2.9 The rotor system configuration with two bearings and two disks. ................. 40

Figure ‎2.10 The Campbell diagram of the rotor system for isotropic bearings (case 1). . 42

Figure ‎2.11 The mode shape of the rotor at 4000 rpm supported by isotropic bearings

(case 1). ............................................................................................................................. 43

Figure ‎2.12 The Campbell diagram of the rotor system supported by anisotropic bearings

(case 2). ............................................................................................................................. 45

Figure ‎2.13: The mode shape of the rotor system supported by anisotropic bearings (case

2). ...................................................................................................................................... 46

Figure ‎2.14: Steady-state response of the rotor-bearing system at node 3 in y and z

directions case 1 (isotropic bearings) ................................................................................ 47

xi


directions for case 1 (isotropic bearings) .......................................................................... 48

Figure 2.16: Steady-state response of the rotor-bearing system at node 3 and node 5 in y

direction for case 1 (isotropic bearings) ............................................................................ 48

Figure 2.17 Campbell diagram of the rotor system supported by isotropic bearings ....... 49

Figure 2.18 The configuration of the overhung rotor with one disk and two bearings .... 50

Figure ‎2.19 The stepped shaft-disk system ....................................................................... 52

Figure 3.1 Single composite material lamina deformed into a uniform cylinder. ............ 57

Figure ‎3.2 Single layer of composite material in the principal material coordinates. ...... 58

Figure ‎3.3 The forces and moments on the cross-section of the composite shaft ............ 65

Figure 3.4 Deformed geometry of an edge of beam under the assumption of Timoshenko

Beam theory ...................................................................................................................... 71

Figure ‎3.5 Campbell diagram of boron/epoxy composite shaft. ....................................... 81

Figure 3.6 Mode shapes of the boron/epoxy composite shaft at 5000 rpm. ..................... 82

Figure ‎3.7 The Campbell diagram of the graphite/epoxy composite shaft-disk system. .. 86

Figure ‎3.8 The mode shapes of the graphite/epoxy composite shaft-disk system at 6000

rpm. ................................................................................................................................... 87

Figure 3.9 Unbalance response and phase angle of the graphite/epoxy composite shaft at

the disk position in y and z directions ............................................................................... 88

Figure ‎3.10 The stepped composite shaft-disk system. ................................................... 89

Figure ‎3.11 The stepped composite shaft under the axial load ......................................... 97

xii

Figure 3.12 The natural frequencies of the stepped composite shaft at 0 rpm under

different axial loads........................................................................................................... 98

Figure 3.13 The backward natural frequencies of the stepped composite shaft at 5000

rpm under different axial loads ......................................................................................... 99

Figure ‎3.14 The forward natural frequencies at 5000 rpm under different axial loads. . 100

Figure ‎4.1 Single composite lamina deformed into tapered cylinder ............................ 103

Figure 4.2 Typical tapered shaft element ........................................................................ 109

Figure ‎4.3 Hierarchical beam finite element with two nodes ......................................... 115

Figure 4.4 Beam element with three nodes ..................................................................... 124

Figure 4.5 The configuration of the tapered graphite - epoxy composite shaft .............. 147

Figure 5.1 The configuration of the tapered composite shaft with disk in the middle ... 152

Figure 5.2 The first critical speeds of the tapered composite shaft for different taper

angles obtained using Hermitian finite element model ................................................... 155


angles determined using Lagrangian finite element model ............................................ 155


angles determined using Hierarchical finite element model ........................................... 156

Figure 5.5 and for the layer of graphite-epoxy with fiber

orientation angle of 0o ..................................................................................................... 156


orientation angle of 90o ................................................................................................... 157

xiii


orientation angle of 45o ................................................................................................... 157


orientation angle of -45o.................................................................................................. 158

Figure ‎5.9 Tapered composite shaft with different positions of the disk. ....................... 159

Figure 5.10 The mode shapes of the tapered composite shaft with taper angle of 0o at

6000 rpm ......................................................................................................................... 163

Figure 5.11 Campbell diagram of the tapered composite shaft with taper angle of 0o. .. 163


6000 rpm ......................................................................................................................... 164

Figure 5.13 Campbell diagram of the tapered composite shaft with taper angle of 3o. . 164


6000 rpm ......................................................................................................................... 165

Figure 5.15 Campbell diagram of the tapered composite shaft with taper angle of 5o. . 165

Figure 5.16 Different lengths of the tapered composite shaft. ........................................ 166

Figure ‎5.17 First critical speeds for different lengths determined using hierarchical finite

element ............................................................................................................................ 171

Figure ‎5.18 First critical speeds for different diameters obtained using hierarchical finite

element ............................................................................................................................ 175

Figure ‎5.19 First critical speeds for different fiber orientation angles based on

hierarchical finite element............................................................................................... 182

xiv

Figure 5.20 First critical speed for different bearing stiffness values determined using

hierarchical finite element............................................................................................... 184

Figure 5.21 First critical speed of the tapered composite shaft with taper angle of 0o for

different axial loads obtained using hierarchical finite element ..................................... 188











Figure ‎5.27 The tapered composite shaft with the attached disk .................................... 191

xv

List of Tables

Table ‎2.1 Eigenvalues and natural frequencies of the rotor supported by isotropic

bearings at two different speeds (case 1). ......................................................................... 41

Table ‎2.2 The first seven critical speeds of the rotor supported by isotropic bearings (case

1). ...................................................................................................................................... 41

Table ‎2.3 Eigenvalues and natural frequencies of the rotor supported by anisotropic

bearings at two different speeds (case 2). ......................................................................... 44

Table ‎2.4 The first seven critical speeds of the rotor supported by anisotropic bearings

(case 2). ............................................................................................................................. 44

Table 2.5 The natural frequencies in Hz under the effect of different tensile and

compression axial loads .................................................................................................... 51

Table ‎2.6 The properties of the stepped shaft-disk system ............................................... 52

Table ‎2.7 The natural frequencies in Hz of the stepped shaft with different lengths L2 .. 53

Table ‎2.8: The natural frequencies in Hz of the stepped shaft with different diameters d2.

........................................................................................................................................... 54

Table ‎3.1 The dimensions of the composite shaft and the properties of the bearing [20]. 79

Table ‎3.2 Properties of the composite materials [20] ....................................................... 79

Table ‎3.3 The first critical speed of the boron-epoxy composite shaft. ............................ 80

Table ‎3.4 The first five critical speeds of the boron-epoxy composite shaft. ................... 81

Table 3.5 The dimensions and properties of the composite shaft-disk system [20] ......... 83

xvi

Table 3.6 The first five critical speeds of the graphite/epoxy composite shaft-disk

system. .............................................................................................................................. 84

Table 3.7 The lowest five eigenvalues in rad/s of the graphite/epoxy composite shaft-disk

system running at 6000 rpm and 0 rpm. ........................................................................... 85

Table ‎3.8 The geometric dimensions and properties of the composite shaft-disk system 90

Table ‎3.9 Natural frequencies in Hz of stepped composite rotor with different lengths .. 91

Table ‎3.10 Natural frequencies in Hz of stepped composite rotor with different fiber

orientation angles .............................................................................................................. 92

Table 3.11 The natural frequencies of the stepped composite shaft-disk system with

different lengths of segment 1 and segment 2 .................................................................. 93

Table ‎3.12 The geometric dimensions of segment 2 of the stepped shaft ........................ 94


different diameters of the segment 2 ................................................................................. 95


different densities of the disk ............................................................................................ 96

Table ‎3.15 The geometric dimensions of the composite shaft-disk system ..................... 97

Table ‎4.1 Properties of the composite materials [20] ..................................................... 147

Table ‎4.2 First critical speed in rpm of the tapered composite shaft with different taper

angles using finite element method and Rayleigh-Ritz method ..................................... 148

xvii

Table 4.3 The natural frequencies in Hz of the tapered composite shaft at 10000 rpm with

different taper angles obtained using finite element method and Rayleigh-Ritz method.

......................................................................................................................................... 149

Table 5.1 The geometric dimensions and properties of the tapered composite shaft ..... 153

Table 5.2 The first critical speed in rpm of the tapered composite shaft for different taper

angles. ............................................................................................................................. 154

Table 5.3 First critical speed in rpm of the tapered composite shaft for different taper

angles and positions of the disk ...................................................................................... 159


angles and stacking sequences obtained using Hermitian finite element model ............ 161


angles and stacking sequences determined using Lagrangian finite element model ...... 161


angles and stacking sequences determined using Hierarchical finite element model .... 162

Table 5.7 Natural frequencies in Hz of the tapered composite shaft with different lengths

......................................................................................................................................... 168

Table 5.8 First critical speed in rpm of the tapered composite shaft with different lengths

and taper angles............................................................................................................... 170

Table ‎5.9 Natural frequencies in Hz of the tapered composite shaft at 5000 rpm for

different diameters obtained using hierarchical finite element. ...................................... 173

xviii

Table ‎5.10 First critical speed in rpm of the tapered composite shaft for different

diameters. ........................................................................................................................ 174

Table 5.11 Natural frequencies in Hz of the tapered composite shaft at 5000 rpm with

different fiber orientation angles obtained using hierarchical finite element - I ............. 177


different fiber orientation angles obtained using hierarchical finite element - II ........... 178

Table ‎5.13 First critical speed in rpm of the tapered composite shaft with different fiber

orientation angles ............................................................................................................ 179


different tensile loads using the hierarchical finite element ........................................... 186


different compressive loads using hierarchical finite element ........................................ 187


different material densities of the disk ............................................................................ 192

Table 5.17 The first critical speed in rpm of the tapered composite shaft for different

material densities of the disk........................................................................................... 193

xix

Nomenclature

Displacement of the shaft in y direction

Displacement of the shaft in z direction

Rotation of the shaft around y direction

Rotation of the shaft around z direction

Rotational speed of the shaft

Mass matrix of the disk

Gyroscopic matrix of the disk

Stiffness of bearing in y direction

Stiffness of bearing in z direction

Damping of bearing in y direction

Damping of bearing in z direction

G Shear modulus

E Young’s modulus

xx

Shape factor depending on the shape of the cross-section and

the Poisson ratio

I

Second moment of area of the cross-section about the neutral

plane

A Area of cross-section of the shaft element

Shear angle in x-z plane

Shear angle in x-y plane

Axial load

Translational mass matrix of uniform metal shaft

Rotational mass matrix of uniform metal shaft

Gyroscopic matrix of uniform metal shaft

Bending stiffness matrix of uniform metal shaft

[ ] Shear stiffness matrix of uniform metal shaft

Geometric stiffness matrix due to axial load of uniform metal

shaft

xxi

Translational shape function matrix based on Euler-Bernoulli

beam theory

Rotational shape function matrix based on Euler-Bernoulli

beam theory

Translational shape function matrix based on Timoshenko beam

theory

Rotational shape function matrix based on Timoshenko beam

theory

Moment of the composite shaft around y axis

Moment of the composite shaft around z axis

Shear force in y direction

Shear force in z direction

[Q] Stiffness matrix of a single lamina

Strain energy of the composite shaft due to bending moments

and shear forces

Strain energy due to axial load

xxii

Total strain energy of the composite shaft

Translational mass matrix of uniform composite shaft

Rotational mass matrix of uniform composite shaft

Gyroscopic matrix of uniform composite shaft

Bending stiffness matrix of uniform composite shaft

Shear stiffness matrix of uniform composite shaft

Geometric stiffness matrix due to axial load of uniform

composite shaft

Transformed stiffness matrix of layer due to fiber orientation

angle ƞ

Transformed stiffness matrix of layer due to taper angle α

Stress matrix corresponding to cylindrical coordinate system

( )

Strain matrix corresponding to cylindrical coordinate system

( )

xxiii

Stress matrix corresponding to coordinate system

Strain matrix corresponding to coordinate system

Shape functions matrix - Hierarchical finite element

formulation

Shape functions - Lagrangian finite element formulation

ƞ Fiber orientation angle

α Taper angle

Stress matrix corresponding to coordinate system

Strain matrix corresponding to coordinate system

Stress Transformation matrix of layer due to angle ƞ

Strain Transformation matrix of layer due to angle ƞ

Stress Transformation matrix of layer due to angle α

Strain Transformation matrix of layer due to angle α

Chapter 1

Introduction and literature survey

1.1 Rotordynamic analysis

Today, rotating machines play an important role in aerospace and power industries. They

are used in applications such as aircraft engines, power plant stations, medical equipment,

automobiles, and helicopters. Designing rotating machines requires determining their

dynamic behavior, so it is essential to use modern techniques to model and analyze the

rotating machines. Studying the dynamic behavior of rotating machines to determine their

dynamic characteristics such as critical speeds, bending natural frequencies and

unbalance response, usually is defined as rotordynamic analysis.

Typically, any rotating machine has three main components: rotor, bearings, and attached

disks or blades. The rotor is considered the heart of rotating machines; and since the

rotating machine interacts with its surroundings, such as a high-pressure fluid or

unbalance of an attached disk or blade, the rotor experiences a high level of vibration.

This unacceptable level of vibration can lead to high stresses in the rotor, noise, a bad

surface finish in machining tools, and damage or failure in bearing or seals.

In fact, vibration of the rotor can happen in different ways: lateral vibration, axial

vibration, and torsional vibration. In lateral vibration, the rotor moves in orbital motion,

which is the result of a combination of motions in horizontal and vertical directions.

Axial vibration happens along the rotor’s axis, while torsional vibration appears as a

twisting in the rotor around its axis. Lateral vibration has a more significant effect on the

2

rotor than axial or torsional vibration, because lateral vibration has more amount of

energy than axial or torsional vibration [1].

Moreover, lateral forces, such as unbalance force that exists in attached disks or blades,

are considered the main reason for lateral vibration. When rotor speed is equal to any

bending natural frequencies of the rotor, the rotor undergoes violent vibration and the

corresponding speed of the rotor is referred to as a critical speed.

1.2 Composite materials

When two or more materials with different properties and mechanical performance come

together to make a new material, the performance and properties of which are designed

to be more advanced than those of the individual materials, the new material is defined as

a composite material [2]. On macroscopic scale, advanced composite materials consist of

two phases: fiber reinforcement material and matrix material. Fiber reinforcement is

stiffer and stronger than the matrix and provides composite materials with high strength

and stiffness, while the matrix holds the fibers in their direction, protects them from

environmental effects, and transfers the load between the fibers [3].

Advanced composite materials have a high specific strength and specific stiffness -to-

weight ratio compared to conventional metallic materials, such as steel and aluminum; in

addition, advanced composite materials have a high fatigue life, good capability to resist

corrosion, and low thermal expansion [3]. As a result of their characteristics, advanced

composite materials are extensively used in the aerospace and automobile industries.

3

Recently, driveshafts in helicopters and vehicles have started to be manufactured with

composite materials rather than conventional metallic materials, because composite

materials improve the dynamic characteristics of the driveshaft. Also, the advanced

composite materials give the driveshaft higher critical speeds and higher bending natural

frequencies compared to conventional metallic materials.

However, an advanced composite shaft cannot transfer the same amount of torque as

conventional metallic shaft that has the same size as the composite shaft; and at certain

level of torsional load, torsional buckling happens in the advanced composite shaft before

the conventional metallic shaft. To overcome this problem, a hybrid metallic composite

shaft can be used; where the composite materials increase the bending natural frequency

and the metallic materials improve the capability of transferring a high amount of torque.

1.3 Finite element method (FEM)

FEM is the most popular computational method, and it is used to solve and analyze a

large range of engineering problems, such as structural dynamics, stress analysis, heat

transfer, and fluid flow. The concept behind FEM is simple. Instead of defining the

approximation functions over the complete domain, as the Rayleigh-Ritz method does,

FEM defines the admissible function only over a finite number of subdomains, named

finite elements. Nowadays, there are many types of computer software based on finite

element method, and the most famous softwares in industry and academics being

ANSYS®, NASTRAN

®, and ABAQUS

®.

4

Finite element method can be divided into two categories: conventional and advanced

finite element method. In conventional finite element method, the rotor usually is

modeled by a beam element that has two nodes in total, one node for each of its ends;

each node has four degrees of freedom, that are two displacements and two rotations.

Since finite element method gives an approximation solution, the rapid convergence of

solutions depends on increasing the number of elements. Today, with fast improvement

in computer capabilities, it is possible to use large number of elements to obtain an

accurate solution. However, for some complex engineering problems, increasing the

number of elements can be expensive and time consuming, even with a sophisticated

computer.

An example of an advanced finite element method is the hierarchical finite element

method. The idea behind the hierarchical finite element is to keep the mesh unchanged

and increase the degree of the admissible functions; this can be achieved by adding

polynomial and trigonometric terms inside the admissible functions to increase the

degrees of freedom that resulted in fast convergence. Moreover, it is possible to obtain a

combination of the conventional finite element and the hierarchical finite element, where

the mesh of the elements and the degree of the admissible functions are changeable; this

combination usually is referred to as an hp- version of finite element method [22].

1.4 Literature survey

Driveshaft design depends on predicting critical speeds and bending natural frequencies.

Different analytical and computational methods have been used to study the behaviour of

5

rotordynamic systems. Herein, the literature survey is divided into two parts. The first

part presents some important rotordynamic analyses of conventional metallic driveshafts.

The second part presents rotordynamic analysis of driveshafts made of composite

materials.

1.4.1 Rotordynamic analysis of conventional metallic shaft

As mentioned before, the finite element method (FEM) is the most popular computational

method, and with advanced computer technology it becomes easier to apply FEM on

complex problems. In fact, there is vast literature on rotordynamic analysis of driveshafts

using FEM.

In his PhD thesis, Ruhl [4] used FEM to model a turbo-rotor system; the model did not

account for rotary inertia, axial load, gyroscopic moments, shear deformation or internal

damping. However, his early investigation, of using FEM for a rotor-bearing system is

considered seminal. Ruhl used his model to predict the instability region and unbalance

response of a rotor-bearing system.

Nelson and McVaugh [5] introduced a procedure to model a rotordynamic system using

FEM; their system consisted of a flexible shaft, rigid disks, and discrete bearings.

Equations of motion of each part of the system (disk, rotor and bearing) were derived

separately and then the general equation of the system was obtained by assembling the

equations of each component (disk, rotor and bearing). The model included the effects of

rotary inertia, gyroscopic moments, and axial load. The finite element had two nodes;

each node had four degrees of freedom, two translations and two rotations.

6

In addition, Nelson and Zorzi [6] studied the effect of a constant axial torque on dynamic

characteristics of a rotor-bearing system, and they found that the critical speeds of the

rotor system were not affected by the level of axial torque. However, when the

rotordynamic system was operated at high speed, the effect of the axial torque on the

stability, reliability and safety cannot be neglected.

Nelson and Zorzi [7] studied the internal damping effect on the dynamic behavior of a

rotor-bearing system. The internal damping was represented in two linear damping forms:

viscous and hysteretic damping. The researchers found that the internal damping had a

negative effect on the stability of system: the viscous damping led the rotor-bearing

system to instability at the first critical speed, while hysteretic damping made the system

unstable at all critical speeds. Nelson [8] used Timoshenko beam theory to include the

effects of transverse shear deformation, gyroscopic moment, rotary inertia, and axial

load. In addition, Chen and Ku [9] used a Ritz finite element technique to predict the

regions of dynamic instability of a rotor-bearing system subjected to periodic axial force;

the rotating shaft was modeled based on Timoshenko beam theory.

Moreover, the tapered shaft captured an interest among rotordynamics researchers.

Greenhill et al. [10] extended the linearly tapered Timoshenko beam theory to develop a

tapered finite element formulation for a shaft; they included the shear deformation in the

finite element as a degree of freedom. The conical element had two nodes and each one

of them had six degrees of freedom: two translations, two rotations, and two shear

deformations. Genta and Gugliotta [11] introduced an axisymmetrical conical beam finite

7

elemeent with two complex degrees of freedom at each node. Also, Khuief and

Mohiuddin [12] developed an equation of coupled bending and torsional motions using

Lagrange’s equation; they derived a conical finite element with two nodes and ten

degrees of freedom for each node. The conical finite element was derived based on

Timoshenko beam theory.

1.4.2 Rotordynamic analysis of composite shaft

Recently, composite materials have been used on a large scale for different structural

applications. Since composite materials have high strength- and stiffness-to-weight ratios,

composite materials have become more attractive to manufacturers in the aerospace and

automobile industries. Advanced composite materials dramatically improve the dynamic

characteristics of shafts in terms of critical speeds, bending natural frequencies, and

unbalance response.

Zinberg and Symonds [13] developed an advanced composite tail rotor for helicopters.

They experimentally studied a boron/epoxy composite shaft and determined its critical

speeds; their results show an improvement in the dynamic behavior of composite shafts

over aluminum shafts. Also, Hetherington et al. [14] experimentally investigated the

dynamic performance of supercritical helicopter power transmission composite shafts;

their experiment indicated that composite shafts can be operated above at least the second

critical speed. Also, it was observed that external damping is required for safe and stable

supercritical operation.

8

Singh and Gupta [15] published formulations for the rotordynamics of composite shafts

by using two different theories: the conventional equivalent modulus beam theory

(EMBT) and a layerwise beam theory (LBT). Performance rotordynamics parameters

such as critical speed, unbalance response, bending natural frequencies, and threshold of

stability were more accurately predicted using LBT than using EMBT. In fact, EMBT has

some limitations that LBT does not. Specifically, EMBT does not measure the effects of

bending-stretching coupling, shear-normal coupling, bending-twisting coupling, cross-

sectional deformations, or out-of-plane warping. In another study, Gubran and Gupta [16]

published a modification of the equivalent modulus beam theory to include the effect of

the stacking sequence and coupling mechanisms of composite materials. The bending

natural frequencies of the composite shaft that was analyzed using the modified EMBT

showed a good agreement with LBT results.

Kim and Bert [17] studied the vibration of cylindrical hollow composite shafts using the

Sanders best first approximation shell theory. They investigated the critical speeds of

different types of composite rotors and compared the results to those obtained using the

classical beam theory, and the compared results well agreed. Hu and Wang [18]

performed vibration analysis on rotating laminated cylindrical shells using the

ABAQUS® finite element program. They studied the influences of rotating speed, end

conditions, shell thickness, shell length, and shell radius on the fundamental bending

natural frequency.

9

Chen and Peng [19] studied the stability of rotating uniform composite shafts. The

composite rotor was studied using the shaft finite element, based on Timoshenko beam

theory, by considering the EMBT. Chang et al. [20] developed a finite element model to

study the dynamic behavior of uniform composite shaft. Their model incorporated rotary

inertia, gyroscopic moments and shear deformation as well as the coupling mechanisms

in composite materials. The shaft was modeled based on a first-order shear deformation

beam theory. Also, Chang et al. [21] performed a vibration analysis of rotating uniform

composite shafts containing randomly oriented reinforcements. Boukhalfa et al. [22]

applied a p-version hierarchical finite element to model a composite shaft; the

hierarchical beam finite element contained two nodes, and each node had six degrees of

freedom. The model included the effects of rotary inertia, gyroscopic moments, and shear

deformation as well as the coupling mechanisms in composite materials, and it was

modeled as Timoshenko beam. In addition, Boukhalfa and Hadjoui [23] published free

vibration analysis of uniform composite shaft using the hp-version of finite element,

which is a combination of the conventional version of finite element (h-version) and the

hierarchical finite element.

Kim Wonsuk, in his PhD dissertation [24], developed a mechanical model for a tapered

composite Timoshenko shaft. The model represented an extended length tool holder at

high speed in end milling or boring operation. The structure of the shaft had clamped –

free supports. Kim used the general Galerkin method to obtain the spatial solutions of the

equations of motion. He studied forced torsion, dynamic instability, forced vibration

10

response, and static strength of a tapered composite shaft subjected to deflection-

dependent cutting forces.

1.5 Objectives of the thesis

Eventhough Kim [24] developed a mechanical model for a tapered composite

Timoshenko shaft and studied forced torsion, dynamic instability, and forced vibration

response of a tapered composite shaft, he did not use the finite element method. In fact

Kim used the general Galerkin method to obtain the spatial solutions of the equations of

motion, and the solution was only under clamped-free condition of the tapered composite

shaft. Based on the author’s knowledge there is no work that has been carried out on the

rotordynamic analysis of tapered composite driveshaft based on the finite element

method. In this thesis, the tapered composite shaft means that the inner and outer

diameters of one end are constant while the inner and outer diameters of the other end

increase with increasing the taper angle.

The objectives of the present thesis are: (1) to develop three different finite element

models: conventional-Hermitian, hierarchical, and conventional-Lagrangian finite

element models for rotordynamic analysis of tapered composite shaft; (2) to investigate

the bending natural frequencies, critical speeds and mode shapes of tapered and uniform

composite shafts; (3) to validate the developed finite element models using the

approximate solution that is obtained based on the Rayleigh-Ritz method; and (4) to

conduct a comprehensive parametric study on uniform and tapered composite

driveshafts.

11

Moreover, rotordynamic analysis of metal driveshaft based on conventional-Hermitian

finite element is very well established, so the conventional-Hermitian finite element is

used for rotordynamic analysis of the uniform and the tapered composite driveshaft in

this thesis. In addition, the conventional-Lagrangian finite element model [20] and

hierarchical finite element model [22] of uniform composite driveshaft have already been

developed; but no work has been done on tapered composite driveshaft using these finite

element methods. Therefore, in the present thesis, the conventional-Hermitian, the

conventional-Lagrangian, and hierarchical finite element models are developed to

establish a comparison between finite element solutions. Furthermore, these finite

element models are validated using the Rayleigh-Ritz method.

1.6 Layout of the thesis

This chapter presents introductory information on rotordynamic analysis, composite

materials, and finite element method. Also, it provides a literature survey on

rotordynamic analysis of conventional metallic driveshafts and composite material

driveshaft.

Chapter 2 presents the rotordynamic analysis of conventional metallic shafts using

conventional finite element.

In chapter 3 a formulation based on conventional-Hermitian finite element is developed

and applied to the rotordynamic analysis of uniform composite driveshafts based on

Timoshenko Beam Theory; the effect of axial load is considered in the formulation.

12

Chapter 4 provides four different formulations for tapered composite driveshafts:

conventional-Hermitian, hierarchical, and Lagrangian finite elements as well as

Rayleigh-Ritz formulation

Chapter 5 gives a detailed parametric study on rotordynamics of tapered composite

shafts. The study includes the effects of stacking sequence, axial load, fiber orientation,

length of the driveshaft and taper angle.

Chapter 6 closes the thesis by providing an overall conclusion of the present work and by

suggesting recommendations for future work.

13

Chapter 2

Rotordynamic Analysis of Rotor-Bearing System Using Conventional

Finite Element Method

2.1 Introduction

In this chapter, vibration analysis of conventional metal driveshaft using conventional

finite element formulation is presented. A finite shaft element is developed for this

purpose. Two different models are developed based on two different beam theories; one

is based on Euler -Bernoulli beam theory and the other model is based on Timoshenko

beam theory. In Euler –Bernoulli beam theory rotary inertia and shear deformation are

ignored, while in Timoshenko beam theory they are considered. Also, the effect of the

axial load is included in both the models. Moreover, rotor-bearing system with tapered

shaft is modeled based on Timoshenko theory. Rotordynamic analysis including

determination of natural frequencies, critical speeds, mode shapes and steady-state

response is conducted and applications to example systems are provided. Rotordynamic

analysis of conventional metal driveshaft is a very well established topic, so the equations

in this chapter are borrowed from References [1, 25, 26] to provide basic information and

introduction about rotordynamic analysis.

2.2 Coordinate system

A typical rotor-bearing system consists of three main parts: rotor, disk and bearing.

Figure 2.1 shows a typical rotor-bearing system. The equations of motion of the rotor-

14

bearing system depend on the equations of motion of each component. Thus, it is

necessary to develop the individual equations of motion of the rotor, bearing and disk.

Then, one can assemble them to obtain general equations of motion for the rotor-bearing

system. Moreover, in order to develop the equations of motion of rotor-bearing system

using finite element method, it is required to divide the rotor into number of elements

which are connected together with nodes. Each element has two nodes located at the ends

where disks and bearings are attached to the rotor. Since the vibration analysis in this

work is focus only on lateral vibration, each node has four degrees of freedom: two

translations in the y and z directions and two rotations about the y- and z-axes. The

translational motions of any node on the rotor are defined from the equilibrium position

by displacements v and w in y and z directions, respectively, while the rotational motions

are defined by rotations and about y- and z-axes, respectively. Here, it is assumed

that the rotations and are small [1].

Figure 2.1 Typical rotor-bearing system

15

In addition, rotor-bearing system is described by two coordinate systems: (a) xyz which is

fixed to the space (b) which is fixed to the cross-section of the disk and the shaft and

rotates with them; the two coordinate systems are related through a set of rotation angles.

The kinetic energy of the shaft element and the disk are function of the instantaneous

angular velocities in frame. So, it is assumed that the instantaneous angular

velocities in xyz frame are about the y-axis,

about the z-axis and about the x-

axis, and they have to be transferred to frame. In order to do this, it is necessary to

choose an order of rotations as it is shown in Fig 2.2, and the rotations are applied in the

following order [1]:

1) about y-axis (it makes x’y’z’ coordinate)

2) about the new z-axis (it makes x”y”z” coordinate)

3) about the new x-axis(it makes coordinate)

So, the instantaneous angular velocities in frame are [1]:

(‎2.1)

16

(‎2.2)

where , and are the instantaneous angular velocities about

respectively.

x

y

z

y,y’

z

x’ x’’, x

y’’

βy

βy

βy

βz

βz

βz

z’,z’’

z

x

y

Figure 2.2 Cross-section rotation angles

17

Furthermore, the general equations of rotor-bearing system can be developed using the

following approach:

a. Establishing the kinetic energy and the strain energy for the element rotor-

bearing system.

b. Developing the lateral displacements and the rotations fields of the rotor using

the finite element method.

c. Using the Lagrange’s Equation to obtain the equation of motion of the shaft

element.

where is the vector of the generalized coordinates, is the vector of

generalized forces .

d. Assembling the individual equations of the shaft element, disks, and bearings to

obtain the general equations of motion of rotor-bearing system.

2.3 Rigid disk element

Energy method is used here to obtain the equation of motion of the disk. The disk is

assumed to be a rigid body made of metallic material as it is illustrated in Fig 2.3, so the

strain energy is neglected and the disk is described only by its kinetic energy. Herein, the

rotational part of the kinetic energy is calculated with respect to frame that is fixed to

the disk. The kinetic energy that results from the translation of the disk is

18

x

y

z

Ω

Figure 2.3 Typical rotating rigid disk.

(‎2.3)

where md is the mass of the disk and and are the translational velocities in y and z

directions. The kinetic energy due to disk rotation is [1]

(‎2.4)

where are the angular velocities of the disk around

,respectively. Ip and Id are the polar moment of inertia and diametral moment of inertia,

respectively. The total kinetic energy is

(‎2.5)

19

(‎2.6)

Since the rotations and are small, sin and cos , and the higher order

terms can be neglected. Thus [1],

(‎2.7)

where

is a constant term and has no effect on the equation of the disk;

represents the gyroscopic (Coriolis) effects of the disk where is the rotation speed and

equals to . The equations of motion of the disk are obtained be applying Lagrange’s

equation to Equation (2.7) [1]:

(‎2.8)

The governing equation of motion for rotating rigid disk is

(‎2.9)

where and are the mass matrix and the gyroscopic matrix of the disk,

respectively.

20

(2.9.a)

(2.9.b)

in equation (2.9) represents the forces and moments terms that act on the disk that

result from mass unbalance, skew position of the disk on the shaft, interconnection

forces and other external effects [1]. In an unbalanced disk, the unbalanced forces come

into play because the mass center of the disk is shifted from its geometric center by an

eccentricity of e and a phase angle of . The unbalance forces in the disk are [1,26]

(‎2.10)

(‎2.11)

Also, the out of balance moments can exist on the shaft. The resulting moments are [1,29]

(‎2.12)

(‎2.13)

where τ is the angle between the center line of the rotor and the axis of rotation, and γ is a

phase angle. So, the force and moment matrix is

(‎2.14)

21

2.4 Bearing element

Mostly, bearings are considered as flexible elements and they are represented in a

mathematical model as springs and dampers as shown in Figure 2.4. The nonlinear

relationship between load and deflection in most categories of bearing makes the analysis

more complex, and to avoid this complexity in rotordynamic analysis one can assume

linear load-deflection relationship of the bearing [1].

K

K

C

C

Figure 2.4 Rotating shaft supported at its ends by two bearings.

The virtual work of the bearing acting on the shaft is [1]

(‎2.15)

Or

(‎2.16)

where and are the components of the generalized force that acts on the shaft by the

bearing [1]

22

(‎2.17)

Equation (2.17) can be written in form

(‎2.18)

Or, in a general form when the damping and stiffness of a bearing are function of the

shaft speed [1] :

(‎2.19)

where

and

2.5 Shaft element

Figure 2.5 shows finite shaft element. The shaft is modeled as a rotating beam element

with distributed mass and stiffness. The element has two nodes located at its ends and

each node has four degrees of freedom; the four degrees of freedom are two translational

displacements in y and z directions and two rotational displacements about y- and z-

axes. Furthermore, the internal displacements of the element are functions of time and the

position along the length of the shaft, and the displacement of any point in the element

can be expressed by the displacements of the end nodes and shape function as [26]

(‎2.20)

23

is the vector of the time-dependent displacements of the nodes in the finite shaft

element.

(‎2.21)

The shape functions ( ) are established by using two different beam

theories, that are Euler-Bernoulli beam theory and Timoshenko beam theory.

y

z

x

v1

v2

w1

w2

βz1

βz2

βy1

βy2

Figure 2.5 Typical finite shaft element

24

2.5.1 Shape functions based on Euler - Bernoulli beam theory

In Euler-Bernoulli beam theory, shear deformation and rotary inertia are ignored, and the

translational displacements and rotational displacements, as shown in Figure 2.6, are

related by [26]

(‎2.22)

(‎2.23)

x

y

βz2

βz1

V(x,t)V1

V2

x

z

βy1

w(x,t)w1 w2

βy2

y-x plane

z-x plane

Figure 2.6 The displacements and the rotations in the two bending planes

25

The deflection within the shaft element in y-x plane is defined by the end displacements

and shape functions. To derive the shape functions, one can consider the y-x plane and

represent the lateral displacement by a cubic polynomial with four parameters

because there are four boundary conditions [1,26] :

(‎2.24)

The translation and rotation boundary conditions in y-x plane are:

(‎2.25)

Applying the lateral and the rotational boundary conditions gives

(‎2.26)

(‎2.27)

(‎2.28)

(‎2.29)

Solving for and , and substituting back into Equation (2.24) gives [1,26]

(‎2.30)

where

( 2.31)

26

where is the non-dimensional parameter:

. Substituting Equation (2.30) into

Equation (2.22) gives [1,26]

( 2.32)

where

( 2.33)

For symmetric shaft, the shape functions for one plane of motion can be used for the

other plane, so the displacement field of the shaft element can be approximated as [26]

(‎2.34)

2.5.2 Shape Functions based on Timoshenko Beam theory

Euler - Bernoulli beam theory is a reasonable approximation for a thin beam, but not for a

thick beam which has two important effects: shear deformation and rotary inertia.

Timoshenko beam theory includes shear effect and rotary inertia. The main concept of

the theory is to remove the assumption that the beam cross-section remains perpendicular

to the beam centerline. Moreover, in Timoshenko beam theory, the rotations of the cross-

27

section centerline consist of two parts: one is caused by the bending and the other one by

the shear deformation [26], as illustrated in Figure 2.7.

(‎2.35)

(‎2.36)

y

x

y-x plane

∂v/∂x

∂v/∂x

xy βz

Figure ‎2.7 Deformed geometry of an edge of beam under the assumption of Timoshenko

beam theory

28

And, the rotation and translation boundary conditions in y-x plane are

(‎2.37)

The shear angle and the lateral displacement should be related together, and to

obtain the relationship between them, the moment equilibrium of the beam must be

considered. The bending moments and the internal shearing forces are [1, 25, 26]

(‎2.38)

where

I: Second moment of area of the cross-section about the neutral plane

A: Area of the cross-section of the shaft element

G: shear modulus

E: Young’s modulus

: Shape factor depending on the shape of the cross-section and the Poisson ratio

For a hollow circular shaft section [1]:

(‎2.39)

where, and for a solid shaft [1]:

29

(‎2.40)

The static equilibrium of the beam in y-x plane can be written as [1, 25, 26]:

(‎2.41)

Substituting Equation (2.35) in Equation (2.41) gives

(‎2.42)

Substituting Equation (2.24) in Equation (2.42) to obtain the shear angle [1]

(‎2.43)

is the shear deformation parameter, and it represents the ratio between bending

stiffness and shear stiffness, and

is the non-dimensional parameter [12]. Now,

substituting the boundary conditions from Equation (2.37) in Equations (2.24) and (2.35)

and finding the four parameters (C0, C1, C2, C3), one can get [1,26]

(‎2.44)

where

(‎2.45)

30

Substituting Equation (2.44) and Equation (2.43) into Equation (2.35) gives [1,26]

(‎2.46)

where

(‎2.47)

The displacement field of the shaft element can be approximated as [26]

(‎2.48)

31

2.5.3 Energy equations

Herein, the equations of motions are obtained based on Timoshenko beam theory only,

and for Euler-Bernoulli beam theory the same procedure can be applied. The kinetic

energy equation of shaft element is similar to that of (kinetic energy equation of) the rigid

disk, so the total kinetic energy for the shaft element may be written as [29,28]

(‎2.49)

where , , are the mass per unit length, diametral moment of inertia, and polar

moment of inertia. The total potential energy of the shaft element, including the elastic

bending energy, shear energy and the energy due to a constant axial load , is [28,29]

(‎2.50)

Substituting Equation (2.48) in Equation (2.49) and Equation (2.50), one can obtain the

kinetic and the potential energies as functions of the displacements of the end nodes and

spatial shape function. The Lagrange’s equation is utilized here to obtain the equation of

motion for free vibration of the rotating shaft element [25,26].

32

(‎2.51)

where

is the translational mass matrix :

(‎2.52)

is the translational mass matrix :

(‎2.53)

is the rotational mass matrix :

(‎2.54)

is the gyroscopic matrix :

(‎2.55)

is the bending stiffness matrix :

(‎2.56)

[ ] is the shear stiffness matrix:

(‎2.57)

is the geometric stiffness matrix due to axial load:

33

(‎2.58)

2.5.4 Tapered driveshaft

There are two methods to model a tapered shaft. First method is to model the tapered

shaft with a large number of uniform shaft elements with different diameters, and the

second method is to incorporate the change in the cross-section of the tapered shaft into

the element definition [10]. The later method is adapted here to obtain the element

matrices based on Timoshenko beam theory. In fact, the procedure to construct the

tapered or conical element is exactly same as producing the uniform shaft element, and

the difference is in the kinetic and strain energy where the cross-section area and inertia

terms must be integrated through the length of the element. Also, the shape functions are

exactly the same as in uniform shaft element except that the shear deformation parameter

becomes [1]

(‎2.59)

where and are the average moment of inertia and the average area of the cross-

section. For rotating tapered shaft element, the element translational mass matrix, the

rotational inertia matrix, the gyroscopic matrix, the bending stiffness matrix, the shear

stiffness matrix and the geometric stiffness matrix due to axial force, respectively are

[25,26].

34

(‎2.60)

(‎2.61)

(‎2.62)

(‎2.63)

(‎2.64)

(‎2.65)

(‎2.66)

2.6 System equations of motion and analysis

The general equation of motion of rotor-bearing system is assembled using Equation

(2.9), Equation (2.18) and Equation (2.51).

(‎2.67)

35

where [M], [C], [G], [K], [Q] and {q} are the mass matrix, damping matrix, gyroscopic

matrix, stiffness matrix, the forcing vector and the system displacement vector of the

rotor-bearing system, respectively.

2.6.1 Whirl speeds analysis

The natural frequencies of the system are determined from the homogeneous form of the

general equations of motion (2.67)

(‎2.68)

where [M], [C] and [K] are real and symmetric matrices and [G] is a real and skew-

symmetric matrix; Equation (2.68) can be written in form[1,26]

(‎2.69)

where the matrix is a positive definite and real symmetric matrix and is an

arbitrary real matrix [1,26]

(‎2.70)

The solution of Equation (2.69) can be assumed in the form

(‎2.71)

Substituting Equation (2.71) into Equation (2.69) yields

(‎2.72)

where

36

(‎2.73)

is an arbitrary real matrix that gives the eigenvalues and the eigenvectors

of the system. The eigenvalues happen in pairs of complex conjugates as well as the

eigenvectors and they have the forms [1,26]

(‎2.74)

(‎2.75)

The real part of the eigenvalues, , represents the damping exponents that are used to

determine the instability region of the driveshaft. The positive values of damping

exponents refer to instability in the driveshaft. In addition, the imaginary parts of the

eigenvalues, , is the whirl speeds or the damped natural frequencies of the system.

2.6.2 Campbell Diagram and Critical Speeds

Campbell diagram is a map of natural frequencies of the driveshaft that shows the

variation of the natural frequencies with the rotation speeds; the Campbell diagram is

used to obtain the critical speeds. Figure 2.8 shows a typical Campbell diagram where the

intersections of the natural frequency curve with the forcing frequency lines represent the

critical speeds. In addition, beside the Campbell diagram, there are two other methods to

obtain the critical speeds; one is called the direct method and the other is the iteration

37

method. The direct method can be used in case that the bearing coefficients are not

function of the rotation speed, but if the bearing coefficients change with rotation speed

the direct method must be replaced by the iteration method [1].

In the iteration method, the first critical speed is assumed as then [M ( )], [K ( )],

[C ( )] and [G ( )] are determined and the first eigenvalue is calculated. A new guess

of the first critical speed is obtained to calculate the first eigenvalue. The process will

continue until acceptable convergence is obtained. In the direct method, the critical

speeds are taken when one of the natural frequencies at a specific speed is equal to the

forcing frequency [1]. The forcing frequency can be written in terms of rotational speed

as

(‎2.76)

where n refers to the level of the lateral force on the shaft. For example, in out of balance

n = 1 and in a four bladed helicopter rotor n = 4. In Equation (2.67) the force is in form

, so the solutions of equation (2.67) will be in form

[1]

(‎2.77)

By putting in equation (2.77), one can get [1]

(‎2.78)

The solution of eigenvalue problem is in complex form; the real part of gives the

critical speed.

38

Figure ‎2.8 Typical Campbell diagram of rotor-bearing system.

2.6.3 Steady-State Synchronous Response

Synchronous force or excitation is defined as the force, whose frequency is similar to

rotor speed. Mostly, the synchronous excitation happens because of the mass unbalance

and disk skew. The equation of motion for the rotor-bearing system is

(‎2.79)

39

where represents the resulting forces and moments of the mass unbalance and the

disk skew [1]

(‎2.80)

(‎2.81)

where represents the real solution part, and represents the force and moment

vector that is acting at the node because of the mass unbalance and the disk skew. The

steady-state solution can be assumed as [1]

(‎2.82)

where is a complex vector. Substituting Equation (2.82) and its derivatives into

Equation (2.79), one can get [1]

(‎2.83)

2.7 Numerical Examples

In this section, examples on dynamic analysis of rotor-bearing system are provided. A

MATLAB program was written to analyze a rotor-bearing system; the program is able to

give the natural frequencies at any speed of the rotor, mode shapes, Campbell diagram,

and critical speeds and unbalance response of the rotor-bearing system.

40

In Figure 2.9, a shaft, with 1.5 m long and 0.05 m diameter, is under study. Two disks are

keyed to the shaft at 0.5 m and 1 m from the left end; the first disk from the left has

diameter of 0.28 m and thickness of 0.07 m while the second disk has diameter of 0.35 m

and thickness of 0.07 m. Also, there are two bearings at each end. The shaft is modeled

be six elements with equal length. The meshing starts form the left side of the shaft, so

the first disk from the left side coincides with node 3 while the second disk coincides

with the node 5. Material properties of the rotor and the disk are: E = 211 GPa; G = 81.2

GPa; Density = 7810 Kg/m3.

a) Case 1 : Isotropic bearings : Kyy = Kzz = 1MN/m

b) Case 2: Anisotropic bearings: Kyy = 1MN/m; Kzz = 0.8MN/m

Figure ‎2.9 The rotor system configuration with two bearings and two disks.

In case 1, stiffness and inertia properties are identical in y-x and z-x planes. As a result,

when the rotor does not rotate the natural frequencies of the rotor happen in pairs as it is

shown in Table 2.1. However, when the rotor starts rotating, the pairs of the natural

frequencies start to separate due to the gyroscopic effect. Figure 2.10 shows the Campbell

41

diagram of the case 1, and one can see the separation of the natural frequencies while the

spinning speed increases.

Table ‎2.1 Eigenvalues and natural frequencies of the rotor supported by isotropic

bearings at two different speeds (case 1).

0 (rpm) 4000 (rpm)

Eigenvalues

( rad/s)

Natural frequency

ωn (Hz)

Eigenvalues

( rad/s)

Natural frequency

ωn (Hz)

0 ± 85.67 i 13.64 0 ± 84.52 i 13.46

0 ± 85.67 i 13.64 0 ± 86.88 i 13.83

0 ± 272 i 43.31 0 ± 249.81 i 39.78

0 ± 272 i 43.31 0 ± 291.89 i 46.48

0 ± 716.5 i 114.09 0 ± 599.78 i 95.51

0 ± 716.5 i 114.09 0 ± 826.77 i 131.65

Table ‎2.2 The first seven critical speeds of the rotor supported by isotropic bearings (case

1).

Critical speeds ( rpm) 816 821 2468 2729 5376 8835 9449

In addition, Campbell diagram can be used to find the critical speeds; the first four

critical speeds from the Campbell diagram are almost same as those were calculated

using the direct method and they are shown in Table 2.2. Furthermore, the mode shapes

of the rotor at 4000 rpm are illustrated in Figure 2.11 and because the stiffness and the

42

inertia are alike in y-x and z-x planes, the orbits in the mode shapes at any point along the

rotor take circular shape.

Figure ‎2.10 The Campbell diagram of the rotor system for isotropic bearings (case 1).

43

Figure ‎2.11 The mode shape of the rotor at 4000 rpm supported by isotropic bearings

(case 1).

In case 2, the bearings on both the ends are anisotropic bearings. At 0 rpm the natural

frequencies do not happen in pairs as in case 1 because stiffnesses in y and z directions

are not identical any more. Table 2.3 shows the natural frequencies of case 2 at 0 rpm and

4000 rpm. Figure 2.12 illustrates the Campbell diagram of the case 2; even though that

the natural frequencies at 0 rpm do not happen in pairs, one can see on the Campbell

Nat Freq = 13.5902Hz Nat Freq = 13.9731Hz



44

diagram that the separation between the natural frequencies still happen as the rotation

speed increases. In addition, under the effect of the anisotropic bearings, in Figure 2.13

the mode shapes form an elliptical orbit rather than a circular orbit as in case 1. Table 2.4

illustrates the critical speeds of the rotor system in case 2.

Table ‎2.3 Eigenvalues and natural frequencies of the rotor supported by anisotropic

bearings at two different speeds (case 2).

0 (rpm) 4000 (rpm)

Eigenvalues

( rad/s)

Natural frequency

ωn (Hz)

Eigenvalues

( rad/s)

Natural frequency

ωn (Hz)

0 ± 81.80 i 13.03 0 ± 81.48 i 12.97

0 ± 85.76 i 13.66 0 ± 85.96 i 13.83

0 ± 252.34 i 40.18 0 ± 237.79 i 37.86

0 ± 271.89 i 43.29 0 ± 284.54 45.31

0 ± 679.22 i 108.16 0 ± 583.11 i 92.85

Table ‎2.4 The first seven critical speeds of the rotor supported by anisotropic bearings

(case 2).

Critical speeds ( rpm) 781 819 2348 2663 5258 8573 9325

45

Figure ‎2.12 The Campbell diagram of the rotor system supported by anisotropic bearings

(case 2).

46

Figure ‎2.13: The mode shape of the rotor system supported by anisotropic bearings (case

2).

In the following example, the steady-state response of the rotor-bearing system of the

previous example is presented. The bearing’s properties are taken as Kyy = Kzz =

0.8MN/m and Cyy = Czz= 80 Ns/m, and the out of balance on the left disk is 0.001 Kg.m.

The response in y and z directions of nodes 3 and 5 are shown in Figure 2.14 and Figure

2.15. The responses in y and z directions are coincident as it is shown on both the figures;

this happens because of the equality of the stiffness in y and z directions. So, for any




47

nodes along the rotor the responses in y and z directions are equal. Also, the comparison

between the responses at node 3 and node 5 in y direction is illustrated in Figure 2.16.

The maximum values of the response of both the nodes happen at the same spinning

speeds which are 792 rpm and 2561 rpm. And, by looking into the Campbell diagram in

Figure 2.17, one can find that the previous spinning speeds are the second and the fourth

critical speeds of the rotor.


directions case 1 (isotropic bearings)

48


directions for case 1 (isotropic bearings)

Figure 2.16: Steady-state response of the rotor-bearing system at node 3 and node 5 in y

direction for case 1 (isotropic bearings)

49

Figure 2.17 Campbell diagram of the rotor system supported by isotropic bearings

50

In the next example, an overhung rotor that is 1.5 m long and with 50 mm diameter is

shown in Figure 2.18. The rotor has at its right end a disk with 350 mm diameter and 70

mm thickness. Also, there are two bearings; one is located at left end of the rotor and the

other is located at 1 m from left end. Six Timoshenko shaft elements with equal lengths

are used here. The natural frequencies of the rotor at 0 rpm and 4000 rpm under the effect

of an axial load are determined. Bearing properties are: Kyy = Kzz = 10 MN/m. Material

properties of the rotor and the disk are : E = 211 GPa; G = 81.2 GPa.

Force

Figure 2.18 The configuration of the overhung rotor with one disk and two bearings

Table 2.5 illustrates the natural frequencies under the effect of the axial load. It is clear

from Table 2.5 that the tensile force increases the natural frequencies while the

compressive load decreases the natural frequencies.

51

Table ‎2.5 The natural frequencies in Hz under the effect of different tensile and

compression axial loads

Speed (rpm) Axial Load

0 KN 10 KN -10 KN 100 KN -100 KN

0

14.35 14.67 14.02 17.23 10.48

14.35 14.67 14.02 17.23 10.48

100.38 100.75 100.01 103.90 100.75

100.38 100.75 100.01 103.90 100.75

132.17 132.42 131.92 134.67 132.42

132.17 132.42 131.92 134.67 132.42

4000

12.13 12.44 11.80 14.99 8.32

16.54 16.84 16.22 19.28 12.85

90.08 90.21 89.95 91.34 88.79

100.94 101.34 100.54 104.75 96.76

103.09 103.55 102.63 107.46 98.33

186.88 187.14 186.61 189.48 184.19

In the last numerical example, a stepped shaft-disk system is shown in Figure 2.19. Two

different cases are presented in this example to determine the natural frequencies. Table

2.6 gives the properties of the stepped shaft-disk system for the two cases. In the first

case the length L1 and the diameters d1 and d2 are fixed, while the length L2 changes with

52

respect to the length L1. Table 2.7 illustrates the natural frequencies of the stepped shaft

with different L2. Clearly, the natural frequencies increase with decreasing L2. In the

second case, the lengths L1 and L2 and the diameter d2 are fixed, while the diameter d2

changes with respect to the diameter d1. The mass, the polar moment of inertia, and the

diametral moment of inertia of the disk are calculated for the first d2 and then used for the

other cases of d2. Table 2.8 shows the natural frequencies when d2 is changed; reducing

the diameter d2 decreases the natural frequencies.

L1, d1 L2, d2

Figure ‎2.19 The stepped shaft-disk system

Table ‎2.6 The properties of the stepped shaft-disk system

Disk

Thickness = 0.01 m Outer Diameter = 0.1 m

Bearings

Kyy = 106 N/m Kzz = 10

6 N/m

Case 1

Length, L1 = 0.06 m Diameter, d1 = 0.01 m Diameter, d2 = 0.005 m

Case 2

Diameter, d1 = 0.01 m Length, L1 = 0.06 m Length, L2 = 0.06 m

53

Table ‎2.7 The natural frequencies in Hz of the stepped shaft with different lengths L2

Speed

(rpm)

L2 = L1 L2 = 0.8 L1 L2 = 0.6 L1 L2= 0.4 L1

0

592.5 714.4 882.6 1131.3

592.5 714.4 882.6 1131.3

1688.5 1978.4 2408.2 3018.9

1688.5 1978.4 2408.2 3018.9

4296 4654.8 5097.3 6016.4

4296 4654.8 5097.3 6016.4

4000

571.2 686.8 846.6 1085.8

613.8 742.5 919.5 1178.4

1651 1945.9 2383.6 3004.4

1729.1 2013.6 2434.8 3034.7

4293.4 4652.5 5094.6 6012.8

4298.8 4657.2 5100.1 6020.2

54

Table ‎2.8: The natural frequencies in Hz of the stepped shaft with different diameters d2.

Speed (rpm) d2 = 0.8d1 d2 = 0.6d1 d2 = 0.4d1 d2= 0.2d1

0

1156.2 792.8 401.6 108

1156.2 792.8 401.6 108

3178.3 2248.6 1140.2 300.7

3178.3 2248.6 1140.2 300.7

5315 4491.8 4158 2598.4

5315 4491.8 4158 2598.4

4000

1143.9 774.5 377.7 82.1

1168.4 811 425.8 135.8

3149.9 2211.6 1104.2 272.8

3207 2287.6 1181.1 348.2

5299.2 4486.6 4156.4 2597.8

5331.4 4497.3 4159.8 2599

2.8 Summary

In this chapter, Euler-Bernoulli beam theory and Timoshenko beam theory were used to

develop finite element models for a conventional metal driveshaft. The effect of axial

load is included in both models. Also, tapered driveshaft is modeled based on

Timoshenko beam theory. Numerical examples are given to perform rotordynamic

analysis and to obtain the natural frequencies and critical speeds of conventional metal

driveshaft.

55

Chapter 3

Rotordynamic Analysis of Uniform Composite Shaft Using Finite

Element Method

3.1 Introduction

Advanced composite materials are increasingly being utilized in a large scale in

mechanical and aerospace applications such as automotive driveshaft and helicopter tail

rotors. Advanced composite materials improve substantially the rotor-dynamic

characteristics of a shaft in terms of critical speeds, bending natural frequencies and

unbalance response. Hence, the composite shaft has all the potentials to replace the

metallic shaft in such applications. In this chapter, the vibration analysis of composite

shaft-disk system is conducted using the conventional finite element formulation. A

composite finite element is developed for this purpose. The strain energy and the kinetic

energy of the composite rotor system are determined to obtain the governing equations of

motion of the system using the Lagrange’s equations. The first-order shear deformable

beam theory is used to establish the strain energy of the composite shaft system, while the

kinetic energy is obtained by considering translational and rotational motions of a moving

coordinate system that is attached to the cross-section of the shaft. In this model, the

Timoshenko beam theory is adopted to include the effects of shear deformation and

rotary inertia. In addition, the effects of gyroscopic forces, axial load, and coupling effect

due to the lamination of composite layers are included. A conventional beam finite

56

element formulation with two end nodes and four degrees of freedom per each node is

used to obtain the mass matrix, the gyroscopic matrix and the stiffness matrix of the

composite rotor system in order to perform the vibration analysis. Using numerical

examples the present model is validated in comparison with the results available in the

literature. Rotor systems with stepped shafts are studied using the developed composite

finite element. The critical speeds, natural frequencies, mode shapes, Campbell diagram

and unbalance response of the stepped composite shaft-disk system are determined.

3.2 Stress - strain relations for a composite material layer

Figure 3.1 illustrates a single lamina deformed into a uniform cylinder where the angle ƞ

represents the angle between fiber orientation in the lamina and x-axis in cylindrical

coordinate system (x, θ, r).The material coordinate of single lamina is depicted in Figure

3.2 where the axes 1 and 2 are the principal material directions. The stress-strain relations

for a lamina in the principal material directions can be expressed as [20]:

(‎3.1)

The above equation may be written as:

(‎3.2)

57

where [Q] is the stiffness matrix of a single lamina, and it is a function of elastic moduli,

shear moduli and the Poisson’s ratio values of the lamina.

y

z

x

θ

ereθ

2

1x

η

eθ

3, er

βy

βz

Figure 3.1 Single composite material lamina deformed into a uniform cylinder.

58

Figure ‎3.2 Single layer of composite material in the principal material coordinates.

To determine the stress-strain relations in cylindrical coordinate system, transformation

from the principal material coordinate system (1, 2, 3) to cylindrical coordinate system

(x, θ, r) should be done. The relation between the stresses in cylindrical coordinate

system (x, θ, r) and principal material coordinate system (1, 2, 3) is [20]

(‎3.3)

where ƞ is the fiber orientation angle, m = cos ƞ, and n = sin ƞ. Equation (3.3) can be

abbreviated as

(‎3.4)

Similarly, one can get the relation between the strains in both the coordinate systems:

1

3

2

59

(‎3.5)

Equation (3.5) can be abbreviated as

(‎3.6)

Using Equations (3.1) - (3.6) with some mathematical manipulations, one can write the

stress-strain relation in cylindrical coordinate system [20] as:

(‎3.7)

The above equation can be written in the following form

(‎3.8)

where [ ] is the transformed stiffness matrix of the layer and can be calculated by the

following equation [20] :

(‎3.9)

60

3.1 Strain-displacement relations

Timoshenko beam theory is considered here to study the transverse vibration of the

uniform composite shaft. So, the displacement fields of the composite shaft are assumed

[20,22] as

(‎3.10)

(‎3.11)

(‎3.12)

where and . , are the displacements of any point of

the composite shaft in x, y and z directions, and are the displacements of a point

on the reference axis of the shaft in y and z directions and and are the rotation

angles of the cross-section about y-axis and z-axis. The strains can be determined using

Equations (3.10) - (3.12) [20] as

(‎3.13)

(‎3.14)

(‎3.15)

(‎3.16)

(‎3.17)

61

(‎3.18)

The strain components in cylindrical coordinate system (x, θ, r) can be written in terms of

the strains in the Cartesian coordinate system (x, y, z) [20] as

(‎3.19)

where and . Substituting Equations (3.13) – (3.18) into Equation

(3.19), one can get [20]

(‎3.20)

(‎3.21)

(‎3.22)

(‎3.23)

(‎3.24)

(‎3.25)

Considering Equations (3.21)-(3.23), one can write Equation (3.7) as

62

(‎3.26)

In Timoshenko beam theory, the shear correction factor is used to adjust the stress

state. Introducing the shear correction factor and choosing its value can be a complex

issue especially for structure such as tapered composite shaft. And, because the objective

of this thesis is to develop finite element models for tapered composite shaft, no attention

is given for the shear correction factor . However, the way that, the correction factor

is introduced and calculated in this thesis for uniform and tapered composite shafts, is

based on [20,27]. Therefore, the Equation (3.26) can be written as

(‎3.27)

3.2 Kinetic and strain energy expressions

The kinetic energy of the composite shaft is analogous to that of the isotropic shaft, so

equation (2.49) is used to obtain:

(‎3.28)

63

where , ,and are the mass per unit length, diametral mass moment of inertia, and

polar mass moment of inertia.

(‎3.29)

(‎3.30)

(‎3.31)

where n is the number of the layers in the laminate, and are the outer radius

and inner radius of the s-th layer; is the material density of the s-th layer. The strain

energy of the composite shaft can be written as:

(‎3.32)

Considering Equation (3.20) – (3.25), Equation (3.32) can be reduced to

(‎3.33)

64

Substituting Equation (3.20), Equation (3.24), and Equation (3.25) into Equation (3.33),

one can obtain the strain energy [20] as

(‎3.34)

From Equation (3.34), one can define the stress couples and the stress resultants [20] as:

(‎3.35)

(‎3.36)

(‎3.37)

(‎3.38)

(‎3.39)

65

(‎3.40)

The forces and moments on the cross-section of the composite shaft are illustrated in

Figure 3.3. Using Equations (3.35) – (3.40) in Equation (3.34), the strain energy can be

written [20] as

(‎3.41)

Where and are the shear force in y and z directions, respectively [20].

(‎3.42)

(‎3.43)

θ

z

y

τxθ τxr

Qy

Qz

My

Mzz

y

x σxx

r

Figure ‎3.3 The forces and moments on the cross-section of the composite shaft

66

Considering Equation (3.27), Equation (3.20), Equation (3.24), and Equation (3.25), one

can write the stress couples and the stress resultant in Equations (3.35) – (3.40) as

(‎3.44)

(‎3.45)

67

(‎3.46)

(‎3.47)

(‎3.48)

68

(‎3.49)

After performing the integration in Equations (3.44) – (3.49), the stress couples and the

stress resultants can be written [20] as:

(‎3.50)

(‎3.51)

(‎3.52)

(‎3.53)

(‎3.54)

(‎3.55)

69

where

(‎3.56)

(‎3.57)

(‎3.58)

(‎3.59)

The strain energy in Equation (3.41) represents the strain energy of the composite

shaft that results from the bending moment and the shear force. In the case of the

composite shaft under a constant axial force, the total strain energy of the composite shaft

is

(‎3.60)

where is the work done on the composite shaft due to a constant axial force P and can

be written [9] as

(‎3.61)

70

3.3 Conventional-Hermitian finite element formulation

In chapter 2, Hermitian finite element formulation was used to develop the equations of

motion of the metal shaft. In this section, Hermitian finite element is used again but to

develop a model for uniform composite shaft. The same procedure that was used to

develop the finite element model for the metal shaft in chapter 2 is followed here to build

up the finite element model for the uniform composite shaft. Timoshenko beam theory is

considered to include the shear deformation; Figure 3.4 shows the relation between the

rotation of cross-section and the shear deformation angle in y-x plane. The element that is

used for the uniform composite shaft has two nodes located at its ends, and each node has

four degrees of freedom: two translations ( , and two rotations ( , ). The relations

between the rotations and the shear angles in y-x plane and z-x plane are

(‎3.62)

(‎3.63)

where and are the shear angles in z-x plane and y-x plane, respectively. To

derive the shape functions, one can consider the y-x plane and represent the lateral

displacement by a cubic polynomial with four parameters:

(‎3.64)

71

y

x

y-x plane

∂v/∂x

∂v/∂x

xy βz

Figure 3.4 Deformed geometry of an edge of beam under the assumption of Timoshenko

Beam theory

The rotation and translation boundary conditions in y-x plane are:

(‎3.65)

(‎3.66)

(‎3.67)

(‎3.68)

The shear angle and the lateral displacement should be related together, and to

obtain the relationship between them, the static equilibrium of the beam must be

considered. The equilibrium of the beam in y-x plane can be written as:

72

(‎3.69)

(‎3.70)

Using Equation (3.62) in Equation (3.70), one can get

(‎3.71)

Now, using Equation (3.63) in Equation (3.71), one can get

(‎3.72)

The shear angles in y-x plane and in z-x plane are constants with axial coordinate x, so

(‎3.73)

Using Equation (3.73) into Equation (3.72), one can obtain the shear angle as:

(‎3.74)

73

has a small influence on the shear angle and can be neglected for simplification,

so the shear angle can be written as

( 3.75)

where

( 3.76)

is the shear deformation parameter for composite shaft, and it represents the ratio

between bending stiffness and shear stiffness. Applying the lateral and rotational

boundary conditions gives

(‎3.77)

(‎3.78)

(‎3.79)

(‎3.80)

Substituting Equation (3.79) in Equation (3.78), one can get

(‎3.81)

From Equation (3.80)

(‎3.82)

Substituting Equation (3.82) in Equation (3.81) gives

74

(‎3.83)

Then, substituting Equation (3.83) in Equation (3.79) and Equation (3.82) to obtain and

as

(‎3.84)

(‎3.85)

Substituting Equations (3.83)-(3.85) and Equation (3.77) in Equation (3.64) to obtain the

lateral displacement as:

(‎3.86)

where

(‎3.87)

(‎3.88)

(‎3.89)

(‎3.90)

(‎3.91)

75

Substituting Equation (3.83) in Equation (3.75), then substituting Equation (3.75) and

Equation (3.86) into Equation (3.62), one can obtain as:

(‎3.92)

where

(‎3.93)

(‎3.94)

(‎3.95)

(‎3.96)

(‎3.97)

To obtain and , the previous procedure for obtaining and

can be repeated again. Consequently, the internal displacements and rotations of the

composite element can be expressed in terms of the displacements and rotations of the

end points and the shape functions as

(‎3.98)

76

Substituting Equation (3.98) into Equation (3.28) the kinetic energy expression and into

Equation (3.60) the strain energy expression and then applying the Lagrange’s equations

the equations of motion of the composite shaft element can be obtained.

Lagrange’s equation :

(‎3.99)

where

(‎3.100)

is the vector of generalized coordinates, and Q is the vector of generalized forces.

Since the composite shaft element has two nodes and each node has four degrees of

freedom, there are eight generalized co-ordinates. And, the generalized co-ordinates for

the shaft element are

( 3.101)

After applying Lagrange’s equation, the equations of motion can be written as

(‎3.102)

where

(‎3.103)

77

(‎3.104)

(‎3.105)

(‎3.106)

(‎3.107)

(‎3.108)

(‎3.109)

(‎3.110)

(‎3.111)

More details on how to obtain Equations (3.103)-(3.111) are given in Appendix A.

3.4 Numerical examples

To validate the conventional-Hermitian finite element model of the uniform composite

shaft, a composite hollow shaft made of boron/epoxy lamina is considered for the

vibration analysis. The configuration geometry and the properties of the composite

78

material are given in Table 3.1 and Table 3.2. A MATLAB® program has been written to

perform the vibration analysis of the composite shaft. The first critical speed of the

boron/epoxy shaft is calculated and compared with those given in the reference papers.

The results of the first critical speed are presented in Table 3.3 and from this table a good

agreement between the results that are based on the present model and that were

predicted using beam theories is observed. In this example, the shaft is modeled by nine

elements of equal length using the conventional-Hermitian finite element model while in

the model of Ref. [20] the shaft was modeled using Lagrangian finite element model by

20 finite elements of equal length. In addition, in Ref. [22] the composite shaft was

modeled using a hierarchical composite finite element and one element with ten

hierarchical terms was enough for convergence.

Moreover, in this example the wall thickness of the composite shaft is relatively small, so

the composite shaft can be considered as a shell structure which is advanced than a beam

structure. Therefore, it can be observed from Table 3.3 that the first critical speed

obtained using Sander’s shell theory [16] is near to the first critical speed obtained

experimentally [12] than those first critical speeds that were calculated using models

based on beam theories. The first five critical speeds of the composite shaft are given in

Table 3.4. The Campbell diagram of the boron/epoxy shaft is presented in Figure 3.5, and

it is clear that the natural frequencies are almost constant for all the rotor spin speeds and

the difference between the frequencies of the forward whirl and the backward whirl is

very small. Figure 3.6 shows the mode shapes of the composite shaft at 5000 rpm, and

79

one can see that the first and second modes are similar and the third and fourth are similar

modes and the fifth and sixth modes also are similar; this indicates that the gyroscopic

effect in this example is not effective even for high frequencies.

Table ‎3.1 The dimensions of the composite shaft and the properties of the bearing [20].

Composite Shaft

Length,

L= 2.47 m

Mean

Diameter, D

= 12.69 cm

Wall thickness,

t = 1.321 mm

Lay-up from

inside

[90/45/-45/06/90]

Shear correction

factor, = 0.503

Bearing

Kyy = 1740 GN/m Kzz = 1740 GN/m

Table ‎3.2 Properties of the composite materials [20]

Properties Boron-epoxy Graphite-epoxy

E11 (GPa) 211 139

E22 (GPa) 24 11

G12=G13 (GPa) 6.9 6.05

G23(GPa) 6.9 3.78

ν12 0.36 0.313

Density (Kg/m3) 1967 1578

80

Table ‎3.3 The first critical speed of the boron-epoxy composite shaft.

Theory or Method

First critical

speed (rpm)

Zinberg and

Symonds [12]

Measured experimentally 6000

Equivalent Modulus Beam Theory 5780

Dos Reis et al.[29]

Bernoulli-Euler beam theory with

stiffness determined by shell finite

elements

4942

Kim and Bert [16]

Sander’s shell theory 5872

Donnell’s shallow shell theory 6399

Bert and Kim [28] Bresse-Timoshenko beam theory 5788

Gupta and Singh

[14]

Equivalent Modulus Beam Theory 5747

Layerwise beam theory (LBT) 5620

Chang et al. [20]

Continuum-based Timoshenko beam

theory

5762

Boukhalfa et al.

[22,23]

Timoshenko beam theory with the p

and hp versions of the finite element

method

5760

The present model

Timoshenko beam theory with

conventional finite element method

5747

81

Table ‎3.4 The first five critical speeds of the boron-epoxy composite shaft.

Critical speed (rpm) 5747 5773 20679 20930 40944

Figure ‎3.5 Campbell diagram of boron/epoxy composite shaft.

0 1000 2000 3000 4000 5000 6000 7000 8000 9000 100000

50

100

150

200

250

300

350

400

Rotor spin speed (rev/min)

Natu

ral freq

uen

cy

(H

z)

First Critical speed5747 rpm

FW1

BW1

BW2

FW2

82

Figure 3.6 Mode shapes of the boron/epoxy composite shaft at 5000 rpm.

In the following example, the frequencies, mode shapes, Campbell diagram and critical

speeds of a graphite/epoxy shaft-disk system are determined. The laminate lay-up of the

shaft is [90/45/-45/06/90] starting from inside; the composite shaft is attached at its center

to a uniform-thickness rigid disk and it is supported at the ends by two identical bearings.

The geometric properties of the composite shaft and the disk are in Table 3.5 as well as

bearing properties. This example is given in References [20,22], and it is given here for

further validation of the conventional-Hermitian finite element model.

94.7794Hz 95.1566Hz

343.2722Hz 344.2892Hz

683.0546Hz 684.4563Hz

BW3

BW2

BW1 FW1

FW2

FW3

83

Table 3.5 The dimensions and properties of the composite shaft-disk system [20]

Composite Shaft

Length,

L = 0.72 m

Inner

diameter

ID = 0.028 m

Outer

diameter

OD = 0.048 m

Lay-up from

inside

[90/45/-45/06/90]

Shear

correction

factor, =

0.503

Disk

Mass,

m = 2.4364 Kg

Diametral mass moment of

inertia, Id = 0.1901 Kg.m2

Polar mass moment of inertia,

Ip = 0.3778 Kg.m2

Bearing

Kyy = 17.5 MN/m Kzz = 17.5 MN/m

Cyy = 500 N.s/m Czz = 500 N.s/m

The mass eccentricity of the disk, e = 5× m.

Figure 3.7 shows the Campbell diagram containing the frequencies of the first two pairs

of bending whirling modes. The intersection points of the line R = 1 (R is a ratio between

the whirling bending frequency and the rotation speed) with the whirling frequency

curves represent the critical speeds of the composite shaft-disk system. In this example,

the shaft is modeled by ten elements of equal length using the conventional-Hermitian

finite element model, but in Ref. [22] the composite shaft was modeled by two elements

and ten hierarchical terms for each element. Unfortunately, no results on this example are

84

given in Ref. [22] except the Campbell diagram. Since the critical speeds can be obtained

from the Campbell diagram, the author of this thesis used the Campbell diagram in Ref.

[22] to obtain the first critical speed which is 7400 rpm. The first five critical speeds of

the composite shaft-disk system determined using the conventional-Hermitian finite

element model are illustrated in Table 3.6.

Table 3.6 The first five critical speeds of the graphite/epoxy composite shaft-disk

system.

Critical speed (rpm) 7294 8685 8700 73033 73702

Moreover, the eigenvalues of the composite shaft running at 6000 rpm and 0 rpm are

listed in Table 3.7, and the agreement in the results can be observed between the

conventional-Hermitian finite element model and Lagrangian finite element model [20].

There is no information given about the meshing in Ref. [20]. The mode shapes of the

composite shaft running at 6000 rpm are illustrated in Figure 3.8. The unbalance response

and phase angle of the shaft at the disk location which is the center of the composite shaft

are shown in Figure 3.9, and it is shown that the response is at its peak at the third critical

speed and this means that the unbalance force excites the forward frequencies. Also,

since the properties of the bearings are identical in z and y directions, the orbits in the

mode shapes at any point along the rotor take circular form and the response in z–x and

y-x planes are same.

85

Table 3.7 The lowest five eigenvalues in rad/s of the graphite/epoxy composite shaft-disk

system running at 6000 rpm and 0 rpm.

Modes

The Present Modal

Chang et al. [20]

Lagrangian finit element

0 rpm 6000 rpm 6000 rpm

1BW 909 909 902.9

1FW 909 910 903.6

2BW 1311 835 826.7

2FW 1311 2057 2048.1

3BW 7779 7767 7841.5

3FW 7779 7794 7868.1

4BW 8149 8148 8215.4

4FW 8149 8151 8218.7

5BW 16205 16192 16305.4

5FW 16205 16217 16332.1

86

Figure ‎3.7 The Campbell diagram of the graphite/epoxy composite shaft-disk system.

87

Figure ‎3.8 The mode shapes of the graphite/epoxy composite shaft-disk system at 6000

rpm.

144.8307Hz 144.9391Hz

133.0053Hz 327.4473Hz

1239.2042Hz 1243.3959Hz

BW3

BW1 FW1

BW2

FW3

FW2

88

Figure 3.9 Unbalance response and phase angle of the graphite/epoxy composite shaft at

the disk position in y and z directions

89

In the next example, a stepped composite shaft-disk system is studied. The effects of the

fiber orientation angle and the shaft’s length on the natural frequencies are studied. The

stepped composite shaft-disk system consists of two segments with different diameters, a

disk of steel at the right end and a bearing at the left end; Figure 3.10 illustrates the

stepped composite shaft-disk system. The shaft is made of boron/epoxy material, and the

geometric properties of the stepped shaft are listed in Table 3.8. The thickness of each

layer is 0.5 mm and length of the first segment L1 is 0.05 m. The properties of the

composite material were already given in Table 3.2.

Figure ‎3.10 The stepped composite shaft-disk system.

Table 3.9 shows the natural frequencies of the stepped composite shaft with different

lengths at different rotational speeds. The length of the segment 1 is fixed while the

length of the segment 2 changes by 20 percent every time. The segment 1 and the

segment 2 of the stepped composite shaft are divided into five elements for each. The

elements of the segment 1 have equal length and the same thing can be said about the

elements of the segment 2. The length of the elements in the segment 2 changes with

changing the length of the segment 2 while the length of the elements of the segment 1 is

always fixed. From Table 3.9, one can see that the natural frequency increases when the

length decreases.

L2

d2

L1

d1

90

Table ‎3.8 The geometric dimensions and properties of the composite shaft-disk system

The Stepped Composite Shaft Properties

Segment Outer Diameter Inner Diameter Lay-up

1 0.015 m 0.005 m [0/90/0/90/0/90/0/90/0/90]

2 0.011 m 0.005 m [0/90/0/90/0/90]

Steel Disk

Density Outer Diameter Inner Diameter Thickness

7810 Kg/m3 0.035 m 0.011 m 0.005 m

Bearing

Kzz = 17.5 MN/m Kyy = 17.5 MN/m

Furthermore, the natural frequencies of the stepped composite shaft with different fiber

orientation angles are presented in Table 3.10. The lengths of the segment 1 and the

segment 2 are each 0.05 m, and the number of layers for segment 1 and segment 2 are 10

and 6, respectively. The fiber orientation angle changes from 0o to 90

o by increment of

30o for all the layers; the natural frequency is maximum when the fiber angle ƞ is 0

o and

then it reduces until it reaches the minimum value when the fiber orientation angle ƞ is

90o.

91

Table ‎3.9 Natural frequencies in Hz of stepped composite rotor with different lengths

Rotational

Speed

(rpm)

Mode L2 = L1

L2 = 0.8

L1

L2 = 0.6

L1

L2 = 0.4

L1

L2 = 0.2

L1

0

BW1 3145.2 3694.6 4399.8 5338.4 6680.7

FW1 3145.2 3694.6 4399.8 5338.4 6680.7

BW2 6461.2 6862.9 7354.8 8080.6 9487.4

FW2 6461.2 6862.9 7354.8 8080.6 9487.4

5000

BW1 3122.1 3665.9 4363.7 5292.3 6623.3

FW1 3168.1 3722.9 4435.4 5384.1 6737.9

BW2 6409.5 6814.2 7312.1 8047.1 9465.3

FW2 6514.2 6913 7399.2 8115.5 9510.6

In the following example, three cases are considered to study the natural frequency of the

stepped composite shaft-disk system. In the first case, the length of segment 1 is

increased and the length of segment 2 is decreased while the total length of the stepped

composite shaft remains the same. In the second case, the diameter of segment 2 reduces

by 20 percent each time. In the last case, the effect of the mass of the disk on the natural

frequencies is determined. The boron/epoxy composite material is used in all the three

cases, and the thickness of each layer is 0.5 mm.

92

Table ‎3.10 Natural frequencies in Hz of stepped composite rotor with different fiber

orientation angles

Rotational

Speed

(rpm)

Mode ƞ = 0o

ƞ = 30o

ƞ = 60o

ƞ = 90o

0

BW1 3978.7 2909.8 2234.3 2132.5

FW1 3978.7 2911.2 2234.5 2132.5

BW2 7515 6505.4 5484.5 4947.1

FW2 7515 6511.3 5485.2 4947.1

5000

BW1 3962.2 2882.2 2203.5 2104

FW1 3994.8 2938.7 2265.3 2160.8

BW2 7456.7 6461.3 5441.1 4901.3

FW2 7574.5 6556.7 5530.1 4994.7

Case 1

The dimensions of the composite shaft-disk system were already given in Table 3.8, and

the total length L of the stepped composite shaft is 10 cm. Five elements of equal length

are used for the segment 1 and the segment 2, and the length of the element in both the

segments changes with changing the segment length. Table 3.11 presents the natural

frequencies of the stepped composite shaft; one can see that when L1 starts to increase

and L2 starts to decrease the natural frequency decreases for all the speeds.

93


different lengths of segment 1 and segment 2

Rotational

Speed

(rpm)

Mode

L1 = 0.5 L

L2 = 0.5 L

L1 = 0.6 L

L2 = 0.4 L

L1 = 0.7 L

L2 = 0.3 L

L1 = 0.8 L

L2 = 0.2 L

L1 = 0.9 L

L2 = 0.1 L

0

BW1 3145.2 3267.4 3454.8 3696.5 3944.8

FW1 3145.2 3267.4 3454.8 3696.5 3944.8

BW2 6461.2 6520 6667.8 7004.0 7838.2

FW2 6461.2 6520 6667.8 7004.0 7838.2

5000

BW1 3122.1 3245.6 3434.7 3679.2 3931.1

FW1 3168.1 3288.9 3474.5 3713.5 3958.2

BW2 6409.5 6465.9 6610.9 6943.9 7775.4

FW2 6514.2 6575.6 6725.9 7065.3 7902.1

94

Case 2

Here, the effect of the diameter of segment 2 on the natural frequency is studied. Table

3.12 lists the properties of segment 2 of the stepped shaft; the properties of segment 1, the

disk and the bearing are same as in case 1. The total length of the stepped shaft is 10 cm

and L1 and L2 are equal. The stepped composite shaft is divided into ten elements with

equal length for all the cases in Table 3.12. The polar mass moment of inertia Ip, the

diametral mass moment of inertia Id, and the mass md of the disk are calculated for the

first case in Table 3.12 then they are used for the other cases in the table. Table 3.13

shows the natural frequencies of the shaft with different diameters of segment 2 at

different rotation speeds. It is clear from Table 3.13 that the natural frequency decreases

when the diameter of segment 2 decreases.

Table ‎3.12 The geometric dimensions of segment 2 of the stepped shaft

Outer Diameter Inner Diameter Lay-up from inside

d2 = d1 0.005 m [0/90/0/90/0/90/0/90/0/90]

d2 = 0.8 d1 0.005 m [0/90/0/90/0/90/0/90]

d2 = 0.6 d1 0.005 m [0/90/0/90/0/90]

d2 = 0.4 d1 0.005 m [0/90/0/90]

d2 = 0.2 d1 0.005 m [0/90]

95


different diameters of the segment 2

Speed

(rpm)

Mode d2 = d1 d2 = 0.8 d1 d2 = 0.6 d1 d2 = 0.4 d1 d2 = 0.2 d1

0

BW1 4147.2 3752.2 3145.2 2330.7 1387

FW1 4147.2 3752.2 3145.2 2330.7 1387

BW2 9890 8221.8 6461.2 4742.1 2961.6

FW2 9890 8221.8 6461.2 4742.1 2961.6

5000

BW1 4136.3 3736.8 3122.1 2296.9 1342.9

FW1 4158.1 3767.4 3168.1 2364.2 1431.3

BW2 9847.6 8168.6 6409.5 4698.1 2926.8

FW2 9932.2 8275.5 6514.2 4788.1 2999.5

Case 3

In this case the effect of the mass of the disk on the natural frequency is studied; the

information about the composite shaft system in Table 3.8 is used here. The total length

of the stepped shaft is 10 cm; the segment 1 and the segment 2 have the same length. The

stepped composite shaft is divided into ten elements of equal length. The results of the

natural frequencies of the stepped shaft are given in Table 3.14; increasing the disk mass

reduces the natural frequency of the stepped composite shaft for all rotation speeds.

96


different densities of the disk

Speed

(rpm)

Mode

Density (Kg / m3)

1000 4000 7000 10000 13000

0

BW1 4165.8 3456.2 3199.9 3011.7 2850.8

FW1 4165.8 3456.2 3199.9 3011.7 2850.8

BW2 10732 80970 6704.2 5964.5 5522.5

FW2 10732 80970 6704.2 5964.5 5522.5

5000

BW1 4161.0 3443.5 3179 2982.6 2814.2

FW1 4170.7 3468.7 3220.5 3040.5 2887.1

BW2 10717 80452 6651.2 5917.1 5481.5

FW2 10747 81492 6758.3 6013.4 5565.3

In the last example, the effect of the axial load on the natural frequency of a composite

shaft is considered. Figure 3.11 shows a stepped composite shaft with two bearings and

one disk, and the required information of the composite shaft system is listed in Table

3.15. The axial force (compressive or tensile) starts from 100 N and then increases by

100 N until it reaches 500 N. The segment 1 of the stepped composite shaft is divided

into five elements of equal length and the same thing applies for the segment 2.

97

Figures 3.12 – 3.14 illustrate the natural frequencies at 0 and 5000 rpm, and it can be

observed that the increase in the tensile force leads to increase in the natural frequencies

for all rotating speeds while increasing the compressive force reduces the natural

frequencies. This is so because the axial tensile force makes the shaft more stiffer thereby

increasing the natural frequencies while the compressive force reduces the shaft stiffness

and results in lower natural frequencies.

Force

L2L1

Figure ‎3.11 The stepped composite shaft under the axial load

Table ‎3.15 The geometric dimensions of the composite shaft-disk system

The Stepped Composite Shaft Properties

Segment Length Outer Diameter Inner Diameter Lay-up

1 L1 = 1 m 0.015 m 0.005 m [0/90/0/90/0/90/0/90/0/90]

2 L2 = 0.5 m 0.011 m 0.005 m [0/90/0/90/0/90]

Disk

Density Outer Diameter Inner Diameter Thickness

7810 Kg/m3 0.105 m 0.011 m 0.005 m

Bearing

Kyy = 30 MN/m Kzz = 30 MN/m

98

Figure 3.12 The natural frequencies of the stepped composite shaft at 0 rpm under

different axial loads

99

Figure 3.13 The backward natural frequencies of the stepped composite shaft at 5000

rpm under different axial loads

100

Figure ‎3.14 The forward natural frequencies at 5000 rpm under different axial loads.

101

3.5 Summary

This chapter presents the vibration analysis of composite shaft using conventional-

Hermitian finite element method. Timoshenko beam theory is adopted, and the effects of

the rotary inertia, shear deformation, gyroscopic forces, axial load, and the coupling

effects due to the lamination of composite layers were considered. The results obtained

using the conventional finite element model were validated with the results available in

the literature and a good agreement has been observed. In addition, a parametric study on

stepped composite shaft with a disk at the end has been carried out. The main conclusions

of this chapter are listed in the following:

1) The natural frequency of the composite shaft is affected significantly by changing

fiber orientation, the rotor’s length, the rotating speed and the axial load.

2) Element with four degrees of freedom per node is good enough to obtain results

that match those given in References [20, 22, 23] which used six degrees of

freedom per node.

3) In stepped composite shaft, when the length of the segment 1 that is thicker and

closer to the bearing is fixed, the natural frequencies of the stepped composite

shaft increase with decreasing the length and increasing the diameter of the

segment 2 that is thinner and connected to the disk.

102

Chapter 4

Rotordynamic Analysis of Tapered Composite Shaft Using Finite

Element Method

4.1 Introduction

One method to improve the dynamic characteristics of a structure is tapering the

structure. For example, tapered beam and tapered plate have better dynamic

characteristics than those of non-tapered beam and plate [24]. Same thing can be said

about the rotating shaft, tapering the shaft can increase the bending stiffness, and this can

lead to an improvement of the dynamic characteristics of the shaft and reduction in

vibration. This chapter provides four computational models for free vibration analysis of

tapered shaft using finite element method and Rayleigh – Ritz method.

In this chapter, a hierarchical composite finite element, a conventional-Lagrangian

composite finite element, a conventional-Hermitian composite finite element, and

Rayleigh-Ritz method are used to model tapered Timoshenko rotating driveshaft; the

effects of taper angle, rotary inertia, gyroscopic forces, axial load, and coupling effect

due to the lamination of composite layers are taken into account.

4.2 Stress-strain relations for tapered cylinder layer

Figure 4.1 shows a single lamina deformed into a conical tube with taper angle α that can

change functionally in x direction. The principal material directions are denoted by 1, 2,

and 3. The axis 1’ extends along the tapered tube surface while 3’-axis is perpendicular to

103

the same surface. The fiber angle ƞ is the angle between 1-axis and 1’-axis and the angle

between 2-axis and 2’-axis.

x

y

z

1

η

2 3,3'

1'

2'

z

y

x

rθ

τxrτxθ

α

σxx

α

α

1'

3'

x

er

2', eθ

1'

Figure ‎4.1 Single composite lamina deformed into tapered cylinder

104

To determine the stress-strain relations of tapered tube lamina in cylindrical coordinate

system, transformation from the principal material coordinate system (1, 2, 3) to

cylindrical coordinate system (x, , r) must be done. To do this, it is necessary to apply

sequence of transformations as following:

1) ƞ about 3-axis in principal material coordinate system (1, 2, 3).

2) α about 2’-axis in primed coordinate system (1’, 2

’, 3

’).

The stress – strain relations for lamina with respect to the principal material coordinate

system were shown in Chapter 3 and they are repeated here again. So, the stress-strain

relations for a lamina in the principal material directions are

(‎4.1)

That is

(‎4.2)

where [Q] is the stiffness matrix of a single lamina, and it is a function of elastic moduli,

shear moduli and the Poisson’s ratio of the lamina. The stresses in primed coordinate

system (1’, 2

’, 3’) are related to the stresses in principal material coordinate system (1, 2,

3) as [24]

105

(‎4.3)

where m = cos ƞ and n = sin ƞ. Equation (4.3) can be abbreviated as

(‎4.4)

Similarly, one can get the relation between the strains in both coordinate systems as:

(‎4.5)


(‎4.6)

Using Equations (4.4) and (4.6) into Equation (4.2), one can get the relation between

stresses and strains in primed coordinate system (1’, 2

’, 3’) as [24]:

=

(‎4.7)

106

The above equation can be written in the following form

(‎4.8)

where ] is the transformed stiffness of the layer and can be calculated by the following

equation

(‎4.9)

Now, the stresses in primed coordinate system (1’, 2

’, 3

’) are related to the stresses in

cylindrical coordinate system (x, , r) as in the following [24]:

(‎4.10)

where c = cos α and s = sin α. Equation (4.10) can be abbreviated as

(‎4.11)

Similarly, one can get the relation between the strains in primed coordinate system (1’, 2

’,

3’) and cylindrical coordinate system (x, , r) as:

(‎4.12)


107

(‎4.13)

Using Equations (4.11) and (4.13) into Equation (4.8), one can write the stress-strain

relation in cylindrical coordinate system (x, , r) as in the following [24]:

=

(‎4.14)

The above equation can be written as

(‎4.15)

where is the transformed stiffness of the layer and can be calculated by the following

equation :

(‎4.16)

Considering Equations (3.21)-(3.23) and introducing the shear correction factor in the

same way as in Equation (3.27), one can write Equation (4.14) as

=

(‎4.17)

108

4.3 Kinetic and potential energy expressions

Since the mass, the diametral mass moment of inertia and the polar mass moment of

inertia of a tapered composite shaft change with respect to the axial coordinate of the

shaft, the kinetic energy for tapered composite shaft is

(‎4.18)

where , , are the mass per unit length, diametral mass moment of inertia, and

polar mass moment of inertia.

( 4.19)

( 4.20)

( 4.21)

where n is the number of the layers in the laminate, and is the density of the layer.

are the outer radius and inner radius of the s-th layer. For conical shape, the

inner radius and outer radius are

109

( 4.22)

( 4.23)

where , , , and are defined in Figure 4.2.

x

ri1

ri2

ro1

ro2

Figure 4.2 Typical tapered shaft element

From Equation (3.41), the strain energy of tapered composite shaft can be written as

( 4.24)

In chapter 3, the stress couples and the stress resultants were defined as

(‎4.25)

(‎4.26)

(‎4.27)

110

(‎4.28)

(‎4.29)

(‎4.30)

Substituting Equation (3.20), Equation (3.24), and Equation (3.25) into Equation (4.17),

and then considering Equation (4.17) for Equations (4.25) - (4.30), one can write the

stress couples and the stress resultants of the tapered composite shaft as

( 4.31)

( 4.32)

111

( 4.33)

( 4.34)

( 4.35)

112

( 4.36)

After applying the integrations in Equations (4.31) – (4.36), the stress resultants and

stress couples of the tapered composite shaft are

( 4.37)

( 4.38)

( 4.39)

( 4.40)

( 4.41)

( 4.42)

where

( 4.43)

113

( 4.44)

( 4.45)

( 4.46)

( 4.47)

( 4.48)

( 4.49)

( 4.50)

( 4.51)

Equations (4.43) - (4.51) represent matrix of the tapered composite shaft. For zero

taper angle and will vanish because . Increasing the taper angle

leads to increasing in and , so the effect of and

will be significant at high

taper angle. Substituting Equations (4.37) – (4.42) into Equation (4.24) gives the strain

energy in terms of matrix as:

114

(‎4.52)

The strain energy Equation (4.52) represents the strain energy of the composite shaft

that results from the bending moment and the shear force, but when the composite shaft is

under a constant axial force, the total strain energy of the composite shaft is

(‎4.53)

115

where is the external work done on the shaft due to a constant axial force P and can be

written as

(‎4.54)

4.4 Finite element formulation

4.4.1 Hierarchical composite shaft element formulation

The shape functions in hierarchical finite element can be established from polynomial or

trigonometric functions. In this work the trigonometric function is chosen to build the

model. Figure 4.3 illustrates a hierarchical beam finite element for tapered composite

shaft. The element has two nodes and each of them have four degrees of freedom (two

translational and two rotational). In hierarchical finite element method, the transverse

displacement field of the beam element in y-direction can be expressed as

(‎4.55)

ξ=0 ξ=1

1 2

L

X,ξ

Figure ‎4.3 Hierarchical beam finite element with two nodes

116

The local coordinate x and non-dimensional coordinate ξ are related by

(‎4.56)

So, Equation (4.55) can be written as

( 4.57)

( 4.58)

( 4.59)

where N is the number of the hierarchical terms of displacement. In local coordinate

system, the nodal displacements in y-direction are

( 4.60)

( 4.61)

117

The displacement in y-direction can be expressed as

( 4.62)

( 4.63)

Substituting Equations (4.59) and (4.61) into Equation (4.63)

(‎4.64)

The shape functions of the displacement can be written as

(‎4.65)

where

( 4.66)

( 4.67)

( 4.68)

where N is the number of the hierarchical terms, n = 1, 2, 3, ...N.

The functions and are polynomial functions and they correspond to the nodal

displacements of the hierarchical element, whilst function is trigonometric function

and it corresponds to the hierarchical terms and contributes only to the internal field of

118

the displacement and does not affect the nodal displacement [22]. Repeating the previous

procedure, one can obtain the shape functions for , and . As a result, the

displacement vector formed by the variables , , and can be written as [22]:

( 4.69)

( 4.70)

( 4.71)

( 4.72)

( 4.73)

( 4.74)

Substituting Equation (4.69) into Equation (4.18), Equation (4.52), and Equation (4.54)

and then applying the Lagrange’s equations, one can get the equations of motion of free

vibration of spinning tapered composite driveshaft. In addition to the number of the

nodes, the number of the generalized co-ordinates depends on the number of the

hierarchical terms. So, the generalized co-ordinates are

119

…….

( 4.75)

…….

…….

…….

where . Also, the generalized co-ordinates can be expressed as

( 4.76)

After applying Lagrange’s equations, the equations of motion of free vibration of rotating

tapered composite driveshaft can be written as

( 4.77)

where

( 4.78)

( 4.79)

120

( 4.80)

( 4.81)

( 4.82)

( 4.83)

( 4.84)

( 4.85)

( 4.86)

( 4.87)

121

( 4.88)

( 4.89)

( 4.90)

( 4.91)

( 4.92)

( 4.93)

( 4.94)

( 4.95)

122

( 4.96)

( 4.97)

( 4.98)

( 4.99)

( 4.100)

( 4.101)

123

( 4.102)

( 4.103)

Appendix B shows in details how to obtain the mass sub-matrices, gyroscopic sub-

matrices and stiffness sub-matrices for tapered composite driveshaft using hierarchical

finite element method.

4.4.2 Lagrangian composite shaft element formulation

The Lagrangian interpolation functions are used here to approximate the displacement

fields of the tapered composite rotor. Figure 4.4 shows an element with three nodes; two

nodes are at the ends of the element and one node is at the center of the element. Each

node has four degrees of freedom, two translational in y and z directions and two

rotational about y and z axes. The displacement fields are approximated by a quadratic

approximation.

( 4.104)

( 4.105)

( 4.106)

( 4.107)

124

1 2 3

L

L/2

Figure 4.4 Beam element with three nodes

By considering the displacement in y-direction, one can write the displacements of the

three nodes in y-direction as [31]

( 4.108)

( 4.109)

( 4.110)

Equations (4.108) – (4.110) can be written in matrix form as

( 4.111)

( 4.112)

where

( 4.113)

( 4.114)

125

( 4.115)

( 4.116)

Substituting Equation (4.112) into Equation (4.104) for , one can get

( 4.117)

where

( 4.118)

( 4.119)

( 4.120)

, , and are the shape functions that are used to approximate field solution of

. Repeating the previous procedure for , , and , one can get same

shape functions as those of . So, the displacement field variables can be written as

( 4.121)

( 4.122)

( 4.123)

126

( 4.124)

Substituting Equations (4.121) - (4.124) into Equation (4.18), Equation (4.52), and

Equation (4.54) and then applying the Lagrange’s equations, one can get the equations of

motion of free vibration of tapered composite driveshaft. After applying Lagrange’s

equations, the equations of motion of the tapered composite shaft based on Lagrangian

composite shaft element formulation can be written as:

( 4.125)

where

( 4.126)

( 4.127)

( 4.128)

( 4.129)

( 4.130)

( 4.131)

127

( 4.132)

( 4.133)

( 4.134)

( 4.135)

( 4.136)

( 4.137)

( 4.138)

( 4.139)

( 4.140)

128

( 4.141)

( 4.142)

( 4.143)

( 4.144)

( 4.145)

( 4.146)

( 4.147)

( 4.148)

( 4.149)

129

( 4.150)

( 4.151)

( 4.152)

( 4.153)

( 4.154)

( 4.155)

( 4.156)

( 4.157)

130

( 4.158)

Appendix C shows in details how to obtain the mass sub-matrices, gyroscopic sub-

matrices and stiffness sub-matrices for tapered composite driveshaft using Lagrangian

composite shaft element formulation

4.4.3 Conventional-Hermitian composite shaft element formulation

In chapter 3, conventional-Hermitian composite shaft element was used to build up

uniform composite shaft element and then to develop the equations of motion of the

uniform composite shaft. The same procedure that was used in section 3.5, to develop the

equations of motion of the uniform composite shaft, is followed here to obtain shaft

element for the tapered composite shaft and then to develop the equations of motion of

the tapered composite shaft. Herein, the tapered composite shaft element has two nodes at

its ends, and each node has four degrees of freedom, two translations ( and two

rotations ( . From section 3.5, and are related to the shear angles in y-x

plane and z-x plane, respectively as:

( 4.159)

( 4.160)

131

To derive the shape functions for the tapered composite shaft element, one can consider

the y-x plane and represent the lateral displacement by a cubic polynomial with

four parameters as:

( 4.161)

The translational and rotational boundary conditions in y-x plane of the tapered shaft

element are

(‎4.162)

(‎4.163)

(‎4.164)

(‎4.165)

The shear angle and the lateral displacement should be related together. To obtain

the relationship between them, the static equilibrium of the beam must be considered.

Thus, the equilibrium of the beam in y-x plane can be written as

( 4.166)

( 4.167)

132

Using Equation (4.159) in Equation (4.167), one can get

(‎4.168)

where ,

, and are the averages of

and ,

respectively. Since matrix changes with axial coordinate x, it is easier to use the

average of matrix in Equation (4.167) to obtain . Now, using Equation (4.160)

in Equation (4.168), one can get

(‎4.169)

The change of the shear angles and along the axial coordinate x of the tapered

composite shaft is assumed to be small, thus

(‎4.170)

Considering Equation (4.170), one can write the shear angle as

133

( 4.171)

The coupling term is neglected for simplification, so the shear angle can be written as

( 4.172)

If one considered the z-x plane, the shear angle is

( 4.173)

Substituting Equation (4.161) into Equation (4.172), the shear angle can be written as

( 4.174)

where

( 4.175)

( 4.176)

Since the shear angle is assumed to be constant along the axial coordinate x of the tapered

composite shaft, one can put . So, the shear angle in Equation (4.174) can be

written as:

( 4.177)

Applying the lateral and rotational boundary conditions gives

( 4.178)

( 4.179)

134

( 4.180)

( 4.181)

From Equation (4.181)

( 4.182)

Substituting Equation (4.182) into Equation (4.180) gives

( 4.183)

To obtain , one can Substitute Equation (4.183) and Equation (4.182) into Equation

(4.179).

( 4.184)

where

( 4.185)

Substituting Equation (4.184) into Equation (4.183) and Equation (4.182), gives

and , respectively.

( 4.186)

( 4.187)

where

( 4.188)

135

( 4.189)

Now, substituting and into Equation (4.161), one can obtain the lateral

displacement as :

( 4.190)

where

( 4.191)

( 4.192)

( 4.193)

( 4.194)

( 4.195)

Substituting Equation (4.190) and Equation (4.172) into Equation (4.159) gives the

rotation as:

( 4.196)

136

where

( 4.197)

( 4.198)

( 4.199)

( 4.200)

If the z-x plane is considered and the previous procedure followed, the internal

displacements and rotations of the tapered element can be expressed in terms of the

displacements and rotations of the end points and the shape functions as:

( 4.201)

137

Equation (4.172) and Equation (4.173) can be written as:

( 4.202)

( 4.203)

Substituting Equation (4.202) and Equation (4.203) into Equation (4.52), the strain

energy can be written as

138

( 4.204)

139

Using Equation (4.201) in Equation (4.204) and Equation (4.54), one can obtain the total

strain energy in Equation (4.53) in terms of the nodal displacements the shape functions,

and the derivatives of the shape functions. Moreover, to obtain the kinetic energy in terms

of the nodal displacements and the shape functions, one needs to substitute Equation

(4.201) into the kinetic energy equation of the tapered composite shaft, Equation (4.18).

Since the total strain energy and the kinetic energy are expressed in terms of the nodal

displacements and the shape functions, one can apply Lagrange’s equations to obtain the

equations of motion. Herein, the generalized co-ordinates for the tapered composite shaft

element are

( 4.205)

After applying Lagrange’s equations, the equations of motion of the tapered composite

driveshaft can be written as:

( 4.206)

where

( 4.207)

( 4.208)

140

( 4.209)

( 4.210)

( 4.211)

( 4.212)

( 4.213)

( 4.214)

( 4.215)

( 4.216)

( 4.217)

141

( 4.218)

( 4.219)

4.5 Rayleigh – Ritz solution

In this section, Rayleigh – Ritz method is utilized to obtain an approximate solution for

the tapered composite shaft. The reason behind using Rayleigh – Ritz method is to

validate the models in section 4.4 which were established using finite element method.

The simply supported condition at the ends of the tapered composite shaft is used to

obtain the model, and the series solution functions are assumed for and in the

form [16]

( 4.220)

( 4.221)

( 4.222)

( 4.223)

where

( 4.224)

142

( 4.225)

( 4.226)

( 4.227)

Here, n is the number of Ritz terms and is whirl frequency. To obtain the equations of

motion, Equations (4.220) - (4.223) must be substituted in Equation (4.18) and Equation

(4.52) which represent the kinetic energy of the tapered composite shaft and the strain

energy of the tapered composite shaft, respectively. After obtaining the energy

expressions in terms of the series solution of and , Lagrange’s equations can

be used to establish the equations of motion of the tapered composite shaft. The equations

of motion of free vibration of the rotating tapered composite shaft are

( 4.228)

where

( 4.229)

143

( 4.230)

( 4.231)

( 4.232)

( 4.233)

( 4.234)

( 4.235)

( 4.236)

( 4.237)

144

( 4.238)

( 4.239)

( 4.240)

( 4.241)

( 4.242)

( 4.243)

( 4.244)

( 4.245)

145

( 4.246)

( 4.247)

( 4.248)

( 4.249)

( 4.250)

( 4.251)

( 4.252)

146

( 4.253)

( 4.254)

4.6 Validation

In the following example, the fundamental natural frequency and first critical speed of the

tapered composite shaft are studied using finite element method and the Rayleigh-Ritz

method. In this analysis, a hollow tapered composite shaft made of graphite-epoxy lamina

is considered. The shaft is simply supported at the ends; the outer diameter of the

composite shaft at the left end is 12.69 cm while the outer diameter at the right end

increases with changing the taper angle. The outer diameter of the composite shaft at the

left end is constant for all the taper angles. Figure 4.5 shows the configuration of the

tapered graphite - epoxy composite shaft. The properties of the graphite-epoxy composite

material are listed in Table 4.1. The wall thickness of the tapered composite shaft is 1.321

mm, and the composite shaft is made of ten layers, each with the same thickness. In

addition, the configuration of the composite shaft is [90o/45

o/-45

o/0

o6/90

o] and the lay-up

starts from the inside. The total length of the composite shaft is 2.47 m. The correction

factor ks for the composite shaft at the zero taper angle is 0.503 [20], yet this value is

considered for all taper angles in this example.

147

Table ‎4.1 Properties of the composite materials [20]

Properties Boron-epoxy Graphite-epoxy

E11 (GPa) 211 139

E22 (GPa) 24 11

G12 = G13 (GPa) 6.9 6.05

G23 (GPa) 6.9 3.78

ν12 0.36 0.313

Density (Kg/m3) 1967 1578

Figure 4.5 The configuration of the tapered graphite - epoxy composite shaft

Six elements, twenty elements, nine elements with equal length are used for hierarchical

finite element, Lagrangian finite element, and Hermitian finite element, respectively; the

same number of the elements was used for all the taper angles in the three models. Also,

in Rayleigh-Ritz method five Ritz terms were used. The first critical speeds of the tapered

composite shaft calculated using finite element method are illustrated in Table 4.2 beside

the speeds obtained using the Rayleigh-Ritz method. It can be seen from Table 4.2 that

148

the difference is small between finite element method and the Rayleigh-Ritz method in

obtaining the first critical speed. And the difference does not exceed 4 percent between

Rayleigh-Ritz method and Lagrangian finite element method in obtaining the first critical

speed when the taper angle is 4o.

Table ‎4.2 First critical speed in rpm of the tapered composite shaft with different taper

angles using finite element method and Rayleigh-Ritz method

Taper

angle,

degrees

Finite element method

Rayleigh-

Ritz method Hierarchical

finite element

Lagrangian finite

element

Hermitian finite

element

0 5220 5219 5220 5220

1 6647 6645 6650 6667

2 7721 7718 7730 7820

3 8531 8525 8548 8772

4 9126 9121 9155 9510

Another observation from Table 4.2 is that the critical speed of the composite shaft

increases when increasing the taper angle, because the circumference of the cross-section

increases through the length of the shaft from the left end to the right end when

increasing the taper angle. This means the amount of composite material increases

through the length of the shaft when increasing the taper angle, which makes the tapered

composite shaft stiffer than the uniform composite shaft. The natural frequencies of the

149

tapered composite shaft are shown in Table 4.3; the results were obtained at 10,000 rpm

and the results obtained by finite element models are comparable with the results

predicted using the Rayleigh-Ritz method.

Table 4.3 The natural frequencies in Hz of the tapered composite shaft at 10000 rpm with

different taper angles obtained using finite element method and Rayleigh-Ritz method.

Taper

angle,

degrees

Mode

Hierarchical

finite element

Lagrangian

finite element

Hermitian

finite element

Rayleigh-

Ritz method

0

BW1 87 86 86 87

FW1 88 87 87 88

BW2 317 316 317 316

FW2 319 319 320 319

1

BW1 111 110 110 110

FW1 112 112 112 112

BW2 387 386 388 386

FW2 390 390 391 389

2

BW1 128 128 128 130

FW1 131 131 131 132

BW2 437 436 439 436

FW2 441 440 443 441

150

3

BW1 142 142 142 146

FW1 146 145 145 149

BW2 473 472 476 473

FW2 477 476 480 477

4

BW1 152 152 152 159

FW1 157 157 157 163

BW2 499 499 504 500

FW2 504 502 508 504

4.7 Summary

Herein, hierarchical finite element, Lagrangian finite element, Hermitian finite element

are used to develop three different finite element models of tapered composite driveshaft

for rotordynamic analysis. These models are based on Timoshenko beam theory, so

rotary inertia and shear deformation are accounted for in these models. In addition, the

effect of axial load, gyroscopic force, coupling due to the lamination of composite layers,

and taper angle are incorporated in the three finite element models. The kinetic energy

and the strain energy of the tapered composite driveshaft were determined and then

Lagrange’s equation was used to obtain the equations of motion. For the purpose of

comparing with the finite element models, Rayleigh-Ritz method was used to develop a

model of tapered composite driveshaft. A numerical example was given to validate the

finite element models, and a good agreement was found between the results of the finite

element models and Rayleigh-Ritz solution.

151

Chapter 5

Parametric Study of Tapered Composite Shaft

5.1 Introduction

In the previous chapter, different mathematical models were established for vibration

analysis of the tapered composite shaft; these mathematical models are the conventional-

Hermitian finite element model, the hierarchical finite element model, and the

Lagrangian finite element model. It is important to assess these models in terms of their

ability to predict the natural frequencies and the critical speeds of the tapered composite

shaft, so in chapter 4 the finite element models were validated using Rayleigh-Ritz

formulation and a good agreement between these models was observed.

Therefore, the conventional-Hermitian finite element model, the hierarchical finite

element model, and the Lagrangian finite element model are credible enough to perform

rotordynamic analysis and to study the effects of different parameters, such as the taper

angle, fiber orientation, and axial load, on the natural frequencies and the critical speeds

of the tapered composite shaft. Two cases of the tapered composite shaft are considered

to perform rotordynamic analysis; the effects of different parameters, such as the taper

angle, fiber orientation, and axial load, on the natural frequencies and the critical speeds

of these two tapered composite shafts are studied in this chapter. In addition, in this

chapter in the analysis of the tapered composite shafts it is assumed that torque buckling

does not happen.

152

5.2 Tapered composite shaft case A

In this section, rotordynamic analysis of the tapered composite shaft is performed using

the conventional-Hermitian finite element model, the hierarchical finite element model,

and the Lagrangian finite element model. The tapered composite shaft has a disk at its

middle and two bearings at the ends; the configuration of the tapered composite shaft is

illustrated in Figure 5.1. The shaft is made of a graphite-epoxy composite material, and

the geometric properties of the composite shaft are given in Table 5.1. Different taper

angles are considered in the analysis. The inner and outer diameters at the left end of the

shaft do not change with changing the taper angle, while at the right end they increase

when increasing the taper angle. The tapered composite shaft is modeled by ten elements,

twenty elements, eight elements of equal length using the conventional-Hermitian finite

element model, the Lagrangian finite element model, and the hierarchical finite element

model, respectively.

Figure 5.1 The configuration of the tapered composite shaft with disk in the middle

153

Table 5.1 The geometric dimensions and properties of the tapered composite shaft

Composite Shaft

Length,

L = 0.72

m

Inner

diameter ,

ID = 0.028

m

Outer

diameter ,

OD = 0.048

m

Lay-up from

inside

[90/45/-45/06/90]

Shear

correction

factor, =

0.56

Disk

Mass,

m = 2.4364 Kg

Diametral mass

moment of inertia, Id =

0.1901 Kg.m2

Polar mass moment of

inertia,

Ip = 0.3778 Kg.m2

Bearing

Kyy = Kzz = 17.5 MN/m Czz = Cyy = 500 N.s/m

Table 5.2 shows the first critical speeds of the tapered composite shaft for different taper

angles, and it can be seen from the table that the first critical speed increases when

increasing the taper angle. However, Figure 5.2 - Figure 5.4 show that the increase in the

first critical speed when increasing the taper angle does not continue because the first

critical speed reaches its maximum at 10o taper angle and then starts to drop off when

increasing the taper angle; to understand why this happens, one needs to return to

Equations (4.43) – (4.51) and to look at Figure 5.5 – Figure 5.8. The equations represent

the matrix that depends on the stiffness and the radius of the layer. Whereas, Figure

154

5.5 – Figure 5.8 represent the material stiffnesses for each single layer of the tapered

composite shaft; from the figures it is clear that is much higher than and

for all the layers and the taper angles, and decreases with increasing the taper

angle except for the layer with fiber orientation of 90o. Consequently, in Figure 5.2 -

Figure 5.4, the inner and the outer radii of the layer control the first critical speed for

taper angle of while controls the first critical speed for taper angle of

.


taper angles.

Taper

angle

Hierarchical finite

element

Lagrangian finite element Hermitian finite element

0o

7295 7328 7328

1o

9710 9772 9760

2o

11467 11551 11537

3o

12820 12912 12903

4o

13855 13935 13935

155


angles obtained using Hermitian finite element model


angles determined using Lagrangian finite element model

0 2 4 6 8 10 12 14 16 18 200.7

0.8

0.9

1

1.1

1.2

1.3

1.4

1.5

1.6x 10

4

Taper angle, Degrees

Fir

st c

riti

cal

speed

, rp

m

0 2 4 6 8 10 12 14 16 18 200.7

0.8

0.9

1

1.1

1.2

1.3

1.4

1.5

1.6x 10

4

Taper angle, degrees

Fir

st c

riti

cal

speed

, rp

m

156


angles determined using Hierarchical finite element model


orientation angle of 0o

0 2 4 6 8 10 12 14 16 18 200.7

0.8

0.9

1

1.1

1.2

1.3

1.4

1.5

1.6x 10

4


Fir

st

cri

tical

sp

eed

, rp

m

157





158


orientation angle of -45o

Moreover, the effect of the disk position on the first critical speed is studied. Figure 5.9

shows the tapered composite shaft with different disk positions. Table 5.3 illustrates the

first critical speed of the tapered composite shaft for different disk positions and taper

angles. For taper angles of 0o

and 1o the maximum value of the first critical speed

happens when the position of the disk is located at the center, while for taper angles

between 2o and 4

o the maximum value of the critical speed happens when the disk is

located at a distance of 4L/10 from the left end. It can be said for high taper angles that

the critical speed reaches its maximum as the disk approaches the left bearing where the

inner and outer diameters are smaller than that at the right end. The results in Table 5.3

are determined using only the conventional-Hermitian finite element model.

159

3L/10

4L/10

5L/10

Figure ‎5.9 Tapered composite shaft with different positions of the disk.

Table 5.3 First critical speed in rpm of the tapered composite shaft for different taper

angles and positions of the disk

Taper angle,

degrees

The position of the disk

3L/10 4L/10 5L/10 6L/10 7L/10

0 5748 6602 7328 6602 5748

1 8144 9570 9760 8889 8159

2 10448 12011 11537 10829 10273

3 12532 13511 12903 12405 12033

4 14224 14408 13935 13639 13440

5 15285 15002 14699 14573 14527

Furthermore, the effect of the stacking sequence of the layers on the first critical speed of

the tapered composite shaft is analyzed. Table 5.4 – Table 5.6 illustrate the first critical

160

speed for different stacking sequences and taper angles. The lay-up for the layers starts

from inside, and there are ten layers with four different fiber orientation angles.

The layers near the outer surface have larger circumferences and volumes than those near

the inner surface of the shaft, and they resist more bending moment than those layers that

near form the inner surface; as a result, the outer surface layers control the stiffness of the

shaft. Consequently, it can be observed from Table 5.4 – Table 5.6 that laying up the

layers that have high stiffness near the outer side of the shaft increases the critical speed.

For example, at a taper angle of 4o, the first critical speed of the configuration [06

o

/90o/45

o/-45

o/90

o] is 13474 rpm, and in this configuration the layers that have fiber

orientation of 0o are laid up on the inner side of the shaft. The layers with fiber

orientation of 0o have higher stiffness than other layers, so laying up them near the outer

surface increases the critical speed. Thus, the configuration [90o/45

o/-45

o/90

o/06

o], where

the layers with 0o fiber orientation are laid-up on the outer side of the shaft, has higher

first critical speed than the other configurations in Table 5.4 – Table 5.5. Moreover, it can

be observed from the Table 5.4 that the difference between the first critical speeds of the

configurations A and E decreases when increasing the taper angle; for example, at 00, 2

o,

and 4o the differences in first critical speeds between the configurations A and E are 19%,

9.4%, and 5.1%, respectively. This is an indication that increasing the taper angle

eliminates to some extent the effect of stacking sequence on the first critical speed and

the natural frequencies.

161


angles and stacking sequences obtained using Hermitian finite element model

Configuration Stacking sequence


0o

1o

2o

3o

4o

A [06o /90

o/45

o/-45

o/90

o] 6475 8949 10832 12324 13474

B [90o/

06o /45

o/-45

o/90

o] 6821 9296 11151 12604 13713

C [90o/45

o/06

o /-45

o/90

o] 7056 9514 11333 12744 13818

D [90o/45

o/-45

o/06

o/90

o] 7328 9760 11537 12903 13931

E [90o/45

o/-45

o/90

o /06

o] 7707 10117 11853 13174 14164


angles and stacking sequences determined using Lagrangian finite element model



0o

1o

2o

3o

4o

A [06o /90

o/45

o/-45

o/90

o] 6475 8962 10848 12333 13478

B [90o/

06o /45

o/-45

o/90

o] 6821 9308 11166 12613 13718

C [90o/45

o/06

o /-45

o/90

o] 7060 9530 11354 12758 13823

D [90o/45

o/-45

o/06

o/90

o] 7328 9772 11551 12912 13935

E [90o/45

o/-45

o/90

o /06

o] 7707 10129 11868 13184 14167

162


angles and stacking sequences determined using Hierarchical finite element model



0o

1o

2o

3o

4o

A [06o /90

o/45

o/-45

o/90

o] 6456 8920 10787 12265 13418

B [90o/

06o /45

o/-45

o/90

o] 6798 9260 11099 12538 13653

C [90o/45

o/06

o /-45

o/90

o] 7032 9477 11278 12676 13751

D [90o/45

o/-45

o/06

o/90

o] 7295 9710 11467 12820 13855

E [90o/45

o/-45

o/90

o /06

o] 7668 10059 11776 13084 14078

In addition, Figure 5.10 – Figure 5.15 illustrate the mode shapes and Campbell diagrams

for the tapered composite shaft with configuration of [90o/45

o/-45

o/06

o/90

o] for three

different taper angles. It can be observed that increasing the taper angle increases the

natural frequency and affects the mode shape. These figures were obtained using the

conventional-Hermitian finite element model

163


6000 rpm

Figure 5.11 Campbell diagram of the tapered composite shaft with taper angle of 0

o.

133.80 Hz 145.41 Hz

145.52 Hz 328.19 Hz

1257.01 Hz 1261.25 Hz

BW1

BW2

BW3

FW1

FW2

FW3

0 0.5 1 1.5 2 2.5

x 104

0

100

200

300

400

500

600


Dam

ped

natu

ral

freq

uen

cie

s (

Hz)

FW2

BW1

FW1

BW2

164

Figure 5.12 The mode shapes of the tapered composite shaft with taper angle of 3

o at

6000 rpm


o.

237.30 Hz 251.25 Hz

344.38 Hz 501.80 Hz

1224.86 Hz 1251.15 Hz

BW2

BW3

BW1FW1

FW2

FW3

0 0.5 1 1.5 2 2.5

x 104

0

100

200

300

400

500

600


Dam

ped n

atu

ral

frequencie

s (

Hz)

BW1

FW2

FW1

BW2

165

Figure 5.14 The mode shapes of the tapered composite shaft with taper angle of 5

o at

6000 rpm


o.

264.9447 Hz 274.7902 Hz

387.4851 Hz 523.5288 Hz

1366.7854 Hz 1420.0371 Hz

BW1 FW1

BW2 FW2

BW3 FW3

0 0.5 1 1.5 2 2.5

x 104

0

100

200

300

400

500

600

700

800

900

1000


Dam

ped

natu

ral

freq

uen

cie

s (

Hz)

BW1

FW1

BW2

FW2

166

5.3 Tapered composite shaft Case B

In the following example, vibration of a tapered composite shaft subjected to different

effects is studied. The tapered composite shaft is fixed by a bearing at one end and is free

at the other end. The shaft is made of boron-epoxy composite material, and the properties

of the composite material are listed in Table 4.1. The tapered composite shaft is made of

ten layers, and the thickness of each layer is 0.25 mm. Also, the length of the shaft L is

0.5 m and the inner diameter di at the free end is 1 cm. The hierarchical finite element

model only is used here, and seven elements of equal length are considered for the

analysis.

L

0.9 L

0.8 L

0.7 L

Figure 5.16 Different lengths of the tapered composite shaft.

167

5.3.1 Effect of length on natural frequencies and first critical speed

In this section, the effect of the length on the natural frequencies and first critical speed of

the tapered composite shaft is discussed. Figure 5.16 shows the configuration of the

tapered composite shaft with different lengths. The length of the tapered composite shaft

changes from L to 0.7 L by 10 percent every time, and the natural frequencies and critical

speeds were obtained for different taper angles for each length. The inner diameter at the

free end of the tapered composite shaft is kept at 1 cm, whereas the inner diameter of the

other end changes with the changing taper angle and length. The stiffness of the bearings

Kyy and Kzz are 10 GN/m. The configuration of the tapered composite shaft is [90o/45

o/-

45o/06

o/90

o] and the lay-up starts from inside. Table 5.7 and Table 5.8 show the natural

frequencies and critical speeds, respectively, of the tapered composite shaft with different

lengths and taper angles. Two rotational speeds, 0 rpm and 5,000 rpm, are considered to

calculate the natural frequencies. One can observe from the tables that the natural

frequencies and first critical speed increase either when the length decreases or when the

taper angle increases.

Furthermore, Table 5.7 shows that, in this example, the gyroscopic effect does not

influence the natural frequency. For instance, the first backward natural frequency at 0

rpm and 5,000 rpm are almost the same for all taper angles. The natural frequencies in

Table 5.7 are obtained using the hierarchical finite element. Moreover, Figure 5.17 shows

the first critical speeds obtained using the hierarchical finite elements. From the figures,

the difference between the first critical speeds increases with an increasing taper angle.

168

Table 5.7 Natural frequencies in Hz of the tapered composite shaft with different lengths

Length,

m

Rotational

speed

(rpm)

Mode Taper angle, degrees

0o

1o

2o

3o

4o

5o

L = 0.5

0

BW1 416 737 1235 1272 1483 1658

FW1 416 737 1235 1272 1483 1658

BW2 1297 2001 2508 2876 3146 3348

FW2 1297 2001 2508 2876 3146 3348

5000

BW1 415 737 1022 1271 1482 1657

FW1 416 738 1024 1273 1484 1660

BW2 1297 2001 2507 2875 3145 3347

FW2 1298 2002 2509 2876 3147 3349

0.9 L

0

BW1 511 864 1175 1444 1672 1860

FW1 511 864 1175 1444 1672 1860

BW2 1582 2335 2871 3258 3542 3754

FW2 1582 2335 2871 3258 3542 3754

5000

BW1 511 864 1175 1444 1671 1859

FW1 512 865 1177 1446 1673 1862

BW2 1582 2335 2870 3257 3541 3753

FW2 1583 2336 2873 3260 3543 3755

169

0.8 L

0

BW1 644 1035 1376 1669 1914 2118

FW1 644 1035 1376 1669 1914 2118

BW2 1970 2774 3342 3748 4045 4269

FW2 1970 2774 3341 3748 4045 4269

5000

BW1 644 1034 1375 1667 1913 2117

FW1 644 1036 1377 1669 1915 2120

BW2 1969 2774 3341 3747 4045 4268

FW2 1971 2775 3343 3750 4047 4270

0.7 L

0

BW1 835 1272 1648 1968 2236 2458

FW1 835 1272 1648 1968 2236 2458

BW2 2514 3371 3969 4395 4707 4941

FW2 2514 3371 3969 4395 4707 4941

5000

BW1 835 1271 1647 1968 2235 2457

FW1 836 1273 1649 1970 2237 2459

BW2 2514 3370 3968 4396 4706 4940

FW2 2516 3372 3970 4397 4708 4942

170

Table 5.8 First critical speed in rpm of the tapered composite shaft with different lengths

and taper angles

Length, m Taper angle

First critical speed based on Hierarchical finite element,

rpm

L = 0.5

0o

24952

1o

44130

2o

61079

3o

75764

4o

88166

5o

98457

0.9 L

0o

30682

1o

51702

2o

70124

3o

85987

4o

99340

5o

110410

0.8 L

0o

38613

1o

61839

2o

82001

3o

99239

4o

113690

171

5o

125650

0.7 L

0o

50021

1o

75917

2o

98155

3o

117100

4o

132740

5o

145730

Figure ‎5.17 First critical speeds for different lengths determined using hierarchical finite

element

172

5.3.2 Effect of shaft diameter on natural frequencies and first critical speed

In this section, the effect of the inner diameter on the natural frequencies and first critical

speed is analyzed. The length of the tapered composite shaft L is fixed at 0.5 m and the

configuration of the tapered composite shaft and the stiffness of the bearing are the same

as in section 5.3.1. To see its influence on the natural frequencies and first critical speed

of the rotating tapered composite shaft, the inner diameter di at the free end is 1 cm and it

is varied form di to 0.7di. The natural frequency results are illustrated in Table 5.9, and it

can be observed that the natural frequency decreases when the inner diameter at the free

end decreases.

In addition, the first critical speeds of the tapered composite shaft are represented in

Table 5.10, and Figure 5.18. These show that reducing the inner diameter at the free end

reduces the critical speed. Also, when the taper angle increases, the difference between

first critical speeds decreases. For instance, when the taper angle is 0o the first critical

speeds of the tapered composite shaft obtained using the hierarchical finite element are

24,952 rpm and 19,427 rpm for inner diameters di and 0.7 di , respectively. But, when the

taper angle is 5o the first critical speeds become 98,457 rpm and 96,671 rpm for di and

0.7 di, respectively. The difference between first critical speed values for 5o is less than

the difference for 0o.

173

Table ‎5.9 Natural frequencies in Hz of the tapered composite shaft at 5000 rpm for

different diameters obtained using hierarchical finite element.

Diameter,

cm

Mode


0o

1o

2o

3o

4o

5o

di =1 cm

BW1 415 737 1022 1271 1482 1657

FW1 416 738 1024 1273 1484 1660

BW2 1297 2001 2507 2875 3145 3347

FW2 1298 2002 2509 2876 3147 3349

0.9 di

BW1 385 709 998 1252 1467 1647

FW1 386 710 1000 1254 1469 1649

BW2 1208 1928 2451 2833 3114 3324

FW2 1209 1930 2453 2835 3116 3327

0.8 di

BW1 354 680 9744 1232 1452 1636

FW1 355 681 975 1234 1455 1639

BW2 1117 1853 2393 2789 3082 3301

FW2 1118 1855 2395 2791 3084 3303

0.7 di

BW1 323 651 950 1213 1438 1624

FW1 323 653 951 1210 1440 1629

BW2 1025 1776 2333 2744 3049 3277

FW2 1026 1778 2335 2746 3051 3280

174

Table ‎5.10 First critical speed in rpm of the tapered composite shaft for different

diameters.

Diameter, cm Taper angle

First critical speed based on Hierarchical

finite element, rpm

di = 1 cm

0o

24952

1o

44130

2o

61079

3o

75764

4o

88166

5o

98457

0.9 di

0o

23118

1o

42453

2o

59661

3o

74636

4o

87313

5o

97597

0.8 di

0o

21275

1o

40697

2o

58234

3o

73511

4o

86471

175

5o

97242

0.7 di

0o

19427

1o

39054

2o

56846

3o

72394

4o

85650

5o

96671

Figure ‎5.18 First critical speeds for different diameters obtained using hierarchical finite

element

176

5.3.3 Effect of fiber orientation on the natural frequencies and first critical speed

In the following example, the influences of ply orientation angle on natural frequencies

and first critical speed of the tapered composite shaft are studied. The configuration and

material properties of the tapered composite shaft from section 5.3.1 are considered. The

ten layers have the same fiber orientation, and the lamination angles vary from 0o to 90

o

to investigate their effects on the natural frequencies and the first critical speed.

Table 5.11 – Table 5.13 present the natural frequency and first critical speed of the

tapered composite shaft with different lamination angles. According to the results in the

tables, the natural frequencies and first critical speed of the tapered composite shaft

decrease with increasing fiber orientation angles of the layers and vice versa. Moreover,

Figure 5.19 shows the first critical speeds that were obtained using hierarchical finite

element. According to the results in Figure 5.19, at 0o taper angle, the first critical speeds

of the tapered composite shaft are close to each other for fiber orientation angle 45o

90o, for example the first critical speed for 45

o and 90

o are 14796 rpm and 13934,

respectively, and the difference between the two first critical speeds is 5.8 %. However

when the taper angle is 5o, the variation between the first critical speeds for fiber

orientation angle 45o 90

o is clearly noticeable where the first critical speed for 45

o

is 70189 rpm and for 90

o is 63428 rpm and the difference between the two critical speeds

is 9.8 %.

177


different fiber orientation angles obtained using hierarchical finite element - I

Fiber orientation

angle

Mode


0o

1o

2o

3o

4o

5o

0o

BW1 469 810 1078 1281 1431 1542

FW1 469 811 1079 1282 1432 1543

BW2 1409 2039 2405 2629 2775 2878

FW2 1410 2039 2405 2629 2776 2879

20o

BW1 316 563 788 988 1162 1312

FW1 316 564 789 990 1165 1315

BW2 1001 1581 2028 2374 2641 2848

FW2 1002 1583 2031 2377 2644 2852

30o

BW1 270 487 692 886 1065 1228

FW1 270 488 694 888 1068 1232

BW2 866 1401 1850 2228 2547 2814

FW2 867 1403 1853 2232 2552 2820

45o

BW1 244 445 640 831 1016 1192

FW1 245 446 642 832 1019 1196

BW2 790 1299 1749 2153 251 2839

FW2 792 1300 1752 2158 2521 2847

178


different fiber orientation angles obtained using hierarchical finite element - II

Fiber

orientation

angle

Mode


0o

1o

2o

3o

4o

5o

60o

BW1 236 432 624 813 999 1179

FW1 237 433 626 816 1002 1183

BW2 764 126 1707 2112 2480 2815

FW2 765 126 1711 2117 2486 2822

70o

BW1 233 426 615 800 980 1154

FW1 234 428 617 803 984 1158

BW2 752 1238 1666 2049 2392 2698

FW2 754 1240 1670 2053 2398 2704

80o

BW1 232 4223 605 782 951 1109

FW1 233 4232 607 785 954 1113

BW2 742 1208 1601 193 2225 2470

FW2 744 1209 1604 194 2229 2475

90o

BW1 232 420 599 769 928 1073

FW1 232 421 601 771 930 1076

BW2 738 1186 1551 1851 2097 2299

FW2 739 1188 1553 1854 2100 2302

179

Table ‎5.13 First critical speed in rpm of the tapered composite shaft with different fiber

orientation angles

Fiber Orientation

angle

Taper angle

First critical speed based on Hierarchical finite

element, rpm

0o

0o

28140

1o

48605

2o

64648

3o

76758

4o

85717

5o

92115

20o

0o

19208

1o

34108

2o

47642

3o

59622

4o

69956

5o

78717

30o

0o

16403

1o

29367

2o

41612

3o

52866

4o

63147

180

5o

72900

45o

0o

14796

1o

26863

2o

38558

3o

49687

4o

60886

5o

70189

60o

0o

14242

1o

25948

2o

37432

3o

48621

4o

59429

5o

69748

70o

0o

14036

1o

25592

2o

36784

3o

47651

4o

58095

5o

68029

181

80o

0o

13938

1o

25300

2o

36138

3o

46493

4o

56269

5o

65390

90o

0o

13934

1o

25188

2o

35785

3o

45749

4o

54983

5o

63428

182

Figure ‎5.19 First critical speeds for different fiber orientation angles based on

hierarchical finite element

183

5.3.4 Effect of the stiffness of the bearing on the first critical speed

This section shows how a bearing’s stiffness can influence the first critical speed of the

tapered composite shaft. This analysis is conducted using the tapered composite shaft

from section 5.3.1. The stiffness of the bearing varies from 0.01 MN/m to 10 GN/m.

Figure 5.20 presents the variation of the first critical speed of the tapered composite shaft

for various levels of bearing stiffness.

The figure shows that, at low bearing stiffness, increasing the taper angle decreases the

first critical speed; despite the fact that, at high bearing stiffness, increasing the taper

angle increases the first critical speed. In addition, it can be observed from the figure that

at a small taper angle the required stiffness for the bearing to be considered as simply

supported condition, which is the condition that increasing the stiffness of the bearing

does not affect the first critical speed and the natural frequencies any more, is lower than

the stiffness required for the bearing at large taper angle.

184

Figure 5.20 First critical speed for different bearing stiffness values determined using

hierarchical finite element.

185

5.3.5 Effect of axial load on natural frequencies and first critical speed

To study the consequence, on the natural frequencies and first critical speed, of applying

axial load, the tapered composite shaft in section 5.3.1 is considered. The tensile and

compressive loads are applied at the free end of the tapered composite shaft, and the

compressive loads are less than the buckling loads. The results of the natural frequencies

and first critical speed of applying the tensile and compressive loads on the tapered

composite shaft are illustrated in Table 5.14, Table 5.15 and Figure 5.21 – Figure 5.26.

The natural frequencies and critical speeds are obtained using the hierarchical finite

element.

According to the results in the tables and figures, the tensile load increases and the

compressive load decreases the natural frequency and critical speed. This is because the

tensile load increases the stiffness of the tapered composite shaft, while the compressive

load decreases it. In addition, increasing the taper angle increases the natural frequency

and the first critical speed for both the tensile and compressive loads.

186


different tensile loads using the hierarchical finite element

Tensile

Load

(KN)

Mode


0o

1o

2o

3o

4o

5o

1

BW1 433 744 1027 1274 1484 1659

FW1 434 746 1029 1276 1486 1662

BW2 1311 2008 2512 2878 3148 3349

FW2 1312 2010 2514 2880 3149 3351

3

BW1 463 759 1037 1281 1489 1664

FW1 464 760 1038 1283 1491 1665

BW2 1339 2022 2522 2886 3154 3354

FW2 1334 2024 2524 2888 3156 3357

6

BW1 506 781 1051 1291 1497 1669

FW1 507 782 1052 1293 1499 1672

BW2 1379 2043 2536 2897 3162 3361

FW2 1380 2045 2538 2899 3166 3364

9

BW1 545 802 1064 1301 1504 1675

FW1 546 803 1066 1303 1507 1677

BW2 1418 2064 2550 2908 3171 3369

FW2 1418 2066 2553 2910 3173 3371

187


different compressive loads using hierarchical finite element

Compressive

Load (KN)

Mode


0o

1o

2o

3o

4o

5o

1

BW1 398 729 1018 1268 1479 1655

FW1 398 731 1019 1270 1481 1658

BW2 1283 1993 2502 2871 3142 3345

FW2 1284 1996 2504 2873 3.144 3346

3

BW1 361 713 1008 1261 1474 1651

FW1 362 715 1006 1263 1477 1654

BW2 1254 1979 2492 2864 3136 3340

FW2 1255 1981 2494 2866 3138 3342

6

BW1 295 688 993 1250 1466 1645

FW1 296 690 995 1253 1468 1648

BW2 1210 1957 2477 2852 3127 3332

FW2 1211 1958 2479 2855 3129 3335

9

BW1 214 662 977 1240 1459 1639

FW1 214 663 979 1242 1461 1641

BW2 1163 1934 2463 2841 3118 3325

FW2 1165 1936 2464 2843 3120 3327

188


different axial loads obtained using hierarchical finite element



189





190





191

5.3.6 Effect of the disk on the natural frequencies and first critical speed

Another factor that can influence the natural frequencies and critical speeds of the tapered

composite shaft is the attached disk. Therefore, this section discusses the influence of the

disk’s mass on the natural frequencies and first critical speeds. The tapered composite

shaft in section 5.3.1 is considered for the analysis, and Figure 5.27 shows the tapered

composite shaft with the attached disk. The disk is attached at the free end of the shaft,

the thickness of the disk is 0.02 m, and the outer and inner diameters of the disk are 0.06

m and 0.015 m, respectively. Table 5.16 and Table 5.17 illustrate the natural frequencies

and first critical speed of the tapered composite shaft for different disk masses, and the

results in Table 5.16 were obtained using the hierarchical finite element. From the tables,

one can observe that increasing the density reduces the natural frequencies and first

critical speed; thus, to eliminate the effect of the disk’s density, increasing the taper angle

can be helpful because increasing the taper angle can increase the natural frequencies and

first critical speed.

Figure ‎5.27 The tapered composite shaft with the attached disk

192


different material densities of the disk

Density

Kg/m3

Mode


0o

1o

2o

3o

4o

5o

1000

BW1 300 491 653 796 920 1029

FW1 302 494 657 800 925 1033

BW2 1039 1555 1931 2209 2418 2575

FW2 1047 1566 1942 2221 2428 2585

4000

BW1 267 421 543 645 729 798

FW1 275 429 552 653 738 807

BW2 928 1347 1650 188 2074 2229

FW2 960 1393 1705 1944 2133 2286

8000

BW1 252 395 507 5978 671 731

FW1 267 411 522 613 686 746

BW2 820 1136 1359 1534 1680 1808

FW2 878 1218 1452 1633 1782 1911

12000

BW1 241 378 484 570 640 696

FW1 262 401 507 593 662 718

BW2 738 997 1177 1319 1439 1545

FW2 812 1090 1278 1425 1548 1656

193


material densities of the disk

Density,

Kg/m3

Taper angle,

degrees

First critical speed based on Hierarchical

finite element, rpm

1000

0 17840

1 29160

2 38847

3 47394

4 54942

5 61590

4000

0 15556

1 24369

2 31422

3 37359

4 42416

5 46746

8000

0 14318

1 22091

2 28189

3 33273

4 37588

194

5 41278

12000

0 13385

1 20435

2 25915

3 30484

4 34385

5 37754

5.4 Summary

In this chapter a comprehensive parametric study of the tapered composite shaft is carry

out. Two cases are considered. In case A the tapered composite shaft with a disk at the

center and with two bearings at the ends, is considered; the hierarchical, the Lagrangian,

and the conventional-Hermitian finite element models are used to study the effects of the

taper angle and stacking sequence on the natural frequencies and first critical speeds.

Furthermore, in case B the effects of the length, diameter, fiber orientation angle, bearing

stiffness, axial load and disk’s mass on the natural frequencies and first critical speeds are

studied; the tapered composite shaft is fixed by bearing at one end and is free at the other

end. In case B the results of the natural frequencies and first critical speeds are

determined using the hierarchical finite element model.

195

Chapter 6

Conclusions, contributions, and future work

6.1 Conclusions

In the present dissertation three finite element models have been developed for

rotordynamic analysis of the tapered composite shaft. These models were developed

using the hierarchical finite element formulation, Lagrangian finite element formulation,

and conventional-Hermitian finite element formulation. The three finite element models

are based on Timoshenko beam theory, and the effects of rotary inertia, transverse shear

deformation, gyroscopic force, axial load, coupling due to the lamination of composite

layers, and taper angle are incorporated in the three finite element models of the tapered

composite shaft.

In order to validate the three finite element models, Rayleigh - Ritz method is used to

obtain an approximate solution for simply supported tapered composite shaft. In chapter

4 a numerical example is given, and it is found that the bending natural frequencies and

first critical speeds, for different taper angles of the tapered composite shaft, determined

using Rayleigh-Ritz method are in agreement with those obtained using the hierarchical,

the Lagrangian, and the conventional-Hermitian finite element models.

In this thesis, the tapered composite shaft means that the inner and outer diameters of

one end are constant while the inner and outer diameters of the other end increase with

increasing the taper angle. Consequently, it is found that increasing the taper angle

196

increases the bending natural frequencies and first critical speed of the tapered composite

shaft. However, it is seen from the numerical results of Case A in chapter 5 that this

direct relationship between the first critical speed and the taper angle does not sustain

because the first critical speed reaches its maximum value at 10o and then starts to drop

off with increasing the taper angle.

In chapter 5 an extensive parametric study of the rotordynamic response of tapered

composite shaft is presented, and the effects of stacking sequence, fiber orientation

angles, taper angle, axial load, bearing stiffness, disk’s position, the inner diameter, and

the length of the tapered composite shaft are studied. The important points that can be

said about the results in chapter 5 are the following:

Stacking the layers that have high stiffness near the outer surface of the shaft

increases the natural frequencies and first critical speed; because the layers near

the outer surface have higher volume and circumference than those near the

inner surface of the shaft.

Increasing the taper angle when using low stiffness bearing decreases the first

critical speed; whereas, increasing the taper angle when using high stiffness

bearing increases the first critical speed.

The natural frequencies and first critical speed of the tapered composite shaft

increase with applying tensile load and decrease with applying compressive

load along the axial coordinate of the tapered composite shaft.

197

Decreasing the length of the tapered composite shaft and increasing the

diameter increase the natural frequencies and the first critical speed and vice

versa.

6.2 Contributions

Using the conventional – Hermitian finite element formulation and following the same

procedure as in References [8,25] to obtain the finite element model for uniform metal

driveshaft, the author of this thesis develops in chapter 3 a finite element model for

uniform composite driveshaft based on Timoshenko beam theory and uses this model to

carry out rotordynamic analysis of the stepped composite shaft.

Three finite element models for tapered composite shaft are developed. These models are

developed using:

1) The hierarchical finite element formulation.

2) The Lagrangian finite element formulation.

3) The conventional-Hermitian finite element formulation.

6.3 Future work

Rotordynamic analysis of the tapered and the uniform composite shafts can be continued,

and the following recommendations can be considered:

Using any one of the three finite element models, dynamic stability of the

tapered composite shaft can be studied.

198

The effect of damping on the response of the tapered composite shaft can be

considered.

The Rayleigh-Ritz model of the tapered composite shaft can be extended to

include the effect of the rigid disk and the bearings rather than considering only

the simply supported condition as it has been done in this thesis.

Manufacturing the tapered composite shaft using the advanced fiber placement

machine and performing experimental rotordynamic analysis.

One of the new applications of the composite material in oil and gas industries

is manufacturing the drill pipe using the composite materials; the drill pipe is

part of the drillstring which is considered as driveshaft. The present work can

be extended to develop a finite element model for drillstring made of composite

materials and to perform the rotordynamic analysis.

199

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[16] H.B.H. Gubran, and K. Gupta, "The effect of stacking sequence and coupling

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204

Appendix A

Uniform Composite shaft-Conventional finite element

The displacement field of the shaft element in terms of nodal displacements and shape

functions is

(A.1)

The kinetic energy of the uniform composite shaft is

(A.2)

The strain energy of the uniform composite shaft due to axial load is

(A.3)

The strain energy of the uniform composite shaft due to bending moments and shear

forces is

205

(A.4)

From chapter 3, the shear angles in y-x plane and z-x plane are

(A.5)

(A.6)

206

Substituting Equations (A.5) and (A.6) into Equation (A.4), one can obtain the following

(A.7)

Also, from chapter 3, the shear angles can be expressed as

(A.8)

(A.9)

where

(A.10)

Substituting Equations (A.8) and (A.9) into Equation (A.7), one can obtain

207

(A.11)

Substituting Equation (A.1) into Equation (A.2), Equation (A.3), and Equation (A.11),

and then applying Lagrange’s equations, one can obtain the equations of motion of the

uniform composite shaft. The generalized co-ordinates for the shaft element are

(A.12)

Using Lagrange’s equations

(A.13)

where

Applying Lagrange’s equation gives

208

(A.14)

(A.15)

(A.16)

209

(A.17)

(A.18)

(A.19)

210

(A.20)

(A.21)

(A.22)

211

(A.23)

(A.24)

212

(A.25)

(A.26)

213

(A.27)

(A.28)

214

(A.29)

215

Appendix B

Hierarchical shaft element formulation

The displacement filed of hierarchical shaft element is [22]

(B.1)

(B.2)

(B.3)

(B.4)

The kinetic energy of the tapered composite shaft in terms of non-dimensional coordinate

is

(B.5)

The strain energy of the tapered composite shaft due to the axial load in terms of non-

dimensional coordinate is

216

(B.6)

The strain energy of tapered composite shaft due to bending moments and shear forces in

term of non-dimensional coordinate is

(B.7)

Substituting Equations (B.1) – (B.4) into Equations (B.5) – (B.7) and then applying

Lagrange’s equations, one can obtain the equations of motion of the tapered composite

shaft.

217

The generalized co-ordinates for the shaft element are

…….

(B.8)

…….

…….

…….

Lagrange’s equation is

(B.9)

where


(B.10)

(B.11)

. .

. .

. .

.

.

.

218

(B.12)

(B.13)

(B.14)

. .

. .

. .

.

.

.

(B.15)

219

(B.16)

(B.17)

. .

. .

. .

.

.

.

220

(B.18)

(B.19)

(B.20)

. .

. .

. .

.

.

.

221

(B.21)

(B.22)

222

(B.23)

. .

. .

. .

.

.

.

223

(B.24)

224

(B.25)

225

(B.26)

. .

. .

. .

.

.

.

226

(B.27)

227

(B.28)

228

(B.29)

. .

. .

. .

.

.

.

229

(B.30)

230

(B.31)

231

(B.32)

. .

. .

. .

.

.

.

232

(B.33)

233

(B.34)

(B.35)

. .

. .

. .

.

.

.

(B.36)

(B.37)

(B.38)

. .

. .

. .

.

.

.

(B.39)

234

Appendix C

Lagrange’s‎interpolation‎formulation

The displacement field for the shaft element with three nodes in terms of nodal

displacements and shape functions is

(C.1)

(C.2)

(C.3)

(C.4)

The kinetic energy of the tapered composite shaft

(C.5)

The strain energy of the tapered composite shaft due to axial load is

235

(C.6)

The strain energy of tapered composite shaft due to bending moments and shear forces is

(C.7)

Substituting Equations (C.1) – (C.4) into Equations (C.5) – (C.7), and then applying

Lagrange’s equations, one can get the equations of motion of the tapered composite shaft.

236

The generalized co-ordinates for the shaft element are

(C.8)

Using Lagrange’s equation

(C.9)

where


(C.10)

(C.11)

(C.12)

237

(C.13)

(C.14)

(C.15)

(C.16)

(C.17)

(C.18)

238

(C.19)

(C.20)

(C.21)

239

(C.22)

(C.23)

(C.24)

240

(C.25)

(C.26)

(C.27)

241

(C.28)

242

(C.29)

243

(C.30)

244

(C.31)

245

(C.32)

246

(C.33)

(C.34)

(C.35)

(C.36)

247

(C.37)

(C.38)

(C.39)

(C.40)

248

Appendix D

Conventional finite element

The displacement field of the shaft element in terms of nodal displacements and shape

functions is

(D.1)

The kinetic energy of tapered composite shaft is

(D.2)

The strain energy of the tapered composite shaft due to axial load is

(D.3)

The strain energy of the tapered composite shaft due to bending moments and shear

forces is

249

(D.4)

250

Substituting Equation (D.1) into Equations (D.2) – (D.4), and then applying Lagrange’s

equations, one can obtain the equations of motion of the tapered composite shaft. The

generalized co-ordinates for the shaft element are

(D.5)

Using Lagrange’s equation

(D.6)

where


(D.7)

251

(D.8)

(D.9)

(D.10)

252

(D.11)

(D.12)

(D.13)

253

(D.14)

254

(D.15)

255

(D.16)

256

(D.17)

257

(D.18)

259

(D.19)

260

(D.20)

261

(D.21)

262

(D.22)

Rotordynamic Analysis of Tapered Composite Driveshaft ......iv and the stiffness matrix of the...

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